







Parents, grandparents, great grandparents, ... 
20180228 

From jessica: every person has 2 birth parents, 4 birth grandparents, and so on. These people are known as ancestors. How many ancestors does a person have in each generation? write in exponential form. 3ed generation, greatgrandparents? 4th generation? 10th generation? Answered by Penny Nom. 





The derivative of f(x)=2^x/x 
20171128 

From Chhavi: f(x)=2^x/x.
Find f'(x). Answered by Penny Nom. 





Average price 
20170915 

From Annette: If I sold a widget at $15.50 17.5% of the time and $16.50 88.5% of the time what is the algebraic expression for that. How do I arrive at an average price Answered by Penny Nom. 





Differentiate y = x^x^x 
20170319 

From Nafis: differentiate y = x^x^x Answered by Penny Nom. 





Renting an electric cart 
20170107 

From Desiree: Vacation Rentals rents electric carts for travel on their grounds. The economy plan rents
carts for $26 per day and the premium plan rents a carts for $12 per day plus a one time fee of $126.05
For how many whole days would it be cheaper to rent from the economy plan then from the premium plan? Answered by Penny Nom. 





The derivative of the inverse of a function 
20161028 

From Kate: Hi, I'm in a College level Calculus course and I can't seem to figure out the answer to this question.
Find the derivative of f^1(4) if f(3)=4 and f'(3)=1/7 Answered by Penny Nom. 





Subtraction of two numbers in different bases 
20161007 

From Wica: Hello,
So happy I found this site. I am having problems with bases. My question is :
Please perform the subtraction between two number over different bases :
A. (5874)base 12  (216)base 9
B. (216)base 9  (5874)base 12 Answered by Penny Nom. 





20/2 + 2 * 2^2 + 6 = 8 
20160915 

From Simone: 20/2 + 2 * 2^2 + 6 = 8
insert brackets to make this true Answered by Penny Nom. 





Implicit differentiation 
20160606 

From Pranay: Is a circle x^2+y^2=2 a function? If it is not a function,
why is it possible to do implicit differentiation on it?
Thanks. Answered by Penny Nom. 





A tangent line to a parabola 
20151202 

From pei: Given that the line y=mx5 is a tangent to the curve y=2x^2+3 find the positive value of M. Answered by Penny Nom. 





The derivative of x^1/3 
20151108 

From omar: hi can help me
am teacher ask me about x^1/3 Derivation definition . Answered by Penny Nom. 





Two cars approaching an intersection 
20150416 

From Engabu: Car A is traveling west at 50km/h & car B is traveling north at 60km/h. both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 3km & car B is 4km from the intersection? Answered by Penny Nom. 





Integers in different bases 
20150305 

From Michael: Let k be a positive integer so that 28 (subscript)k = 132 (subscript)5 Answered by Penny Nom. 





f(x)=(x^21)/(x1) 
20150221 

From Ahmed: Is f(x)=[(x^21)/(x1) and x=2 at x=1] differentiable at x=1 ? Why ? Answered by Penny Nom. 





Differentiate ln[x(2x4)^1/2] 
20140628 

From Igwe: If y=In[x(2x4)^1/2],find dy/dx at x=3 Answered by Penny Nom. 





A tangent of the curve (x/a)^n+(y/b)^n =2 
20140415 

From sudhir: the equation of tangent of the curve (x/a)^n+(y/b)^n =2. at(a,b) is Answered by Penny Nom. 





A number with all the digits different 
20140207 

From Jon: I'm not a mathematician but just curious. Is there a name for a number of up to ten digits where each digit is different,
i.e the equivalent of an isogram in linguistics? My online banking gives me a random 8digit number each time I log on
and it is rare to get one of the type I refer to. There must be calculable odds, but I'm only allowed one question!
Thanks in advance
Jon Answered by Harley Weston. 





Three roommates share the rent 
20131123 

From Cortlin: Because of different bedroom sizes, three roommates decide to split the rent this way:
Roommate A pays 75% as much as B does, and C pays 40% as much as B does.
What fractional part and percent of the rent does each roommate pay? Answered by Penny Nom. 





Differentiate x^x  2^sinx 
20130809 

From tarun: derivative of x^x  2^sinx Answered by Penny Nom. 





Sharing in a car rental 
20130409 

From Marilyn: 8 persons are planning to share equally the cost of a rental car.
If 1 person withdraws from the arrangement and the others share
equally the entire cost of the car, then the share of each of the
remaining persons is increased by? Answered by Penny Nom. 





Driving a rental car 
20130313 

From teresa: Driving a rental car x miles cost y= 0.25x + 25 dollars.
(a) How much would it cost to rent the car but not drive it?
(b) How much does it cost to drive the car 1 additional mile?
(c) What is the yintercept of y= 0.25x + 25? What does it represent?
(d) What is the slope of y= 0.25x + 25? What does it represent? Answered by Penny Nom. 





f(x) + f ''''(x)=0 
20130305 

From Andreea: Hei. I don’t speak lot of english but here is my question,hope u understand: f(x) + f ````(x)=0. so, my question. what is f(x), where f ````(x) is f(x) derivative by four time ? i tried to find the answer and i knew f(x) is something like that f(x)=e^x*sinx but miss something. Answered by Brennan Yaremko. 





The multiplication table for the different bases 
20130201 

From sylvia: I am having a difficult time trying to figure out how to fill in the multiplication table for the different bases. i don't know how to get the numbers. Answered by Penny Nom. 





The derivative of y = sin (30º + x) 
20121107 

From Saskia: derivative of y = sin (30º + x) Answered by Harley Weston. 





An implicit differentiation problem 
20121026 

From Katie: find y' of x^2y2y^3=3x+2y Answered by Harley Weston. 





Differentiation rules 
20121023 

From Morgan: Use the derivative rules to differentiate each of the following:
1. f(x)=1/x1 2. f(x)= sqrt(x) Answered by Penny Nom. 





The derivative of 2sin cubed x  3 sin x 
20120325 

From holly: suppose f(x) = 2sin cubed x  3 sin x
show that f 1(x) = 3 cos x cos 2x Answered by Harley Weston. 





Integral 1/(25x^2)^3/2 
20120222 

From John: Integral 1/(25x^2)^3/2 Answered by Harley Weston. 





Dt[sin t tan (t^2+1)] 
20120221 

From Ayu: Ayu
Dt[sin t tan (t^2+1)]
derivatives Answered by Harley Weston. 





Notation for the second derivative 
20120206 

From Shafira: In all math textbooks, it is written that d/dx ( d/dx) (y)= d2y/dx2. Why do they write it as d2y/dx2, not as d2y/d2x2? Answered by Robert Dawson. 





The derivative of x^(1/2) 
20120114 

From Eric: I have an problem figuring out the derivative of the negative square root of x i.e. x^(1/2) using the first principle.
Could someone please show me?
Thanks in advance! Answered by Harley Weston. 





The number of digits in a number base m 
20111222 

From Jash: Assume there is a number system of base m.
The one property of this system is: If 2 numbers written in this system, which have 'a' and 'b' as the number of digits are multiplied, then the product of the 2 numbers will have a number of digits which is a function f(a,b).
In other words, as long as the number of digits of the 2 numbers are constant, the number of digits of their product is a constant.
Find m. Answered by Robert Dawson. 





Implicit differentiation 
20111020 

From Monica: Find dy/dx in terms of x and y, if sin(xy)=(x^2)y. Answered by Penny Nom. 





The derivative of f(x) = (x+1)^1/2 
20110905 

From Carla: Find the derivative using the limit process of
f(x) = (x+1)^1/2 Answered by Harley Weston. 





Differentiable on an interval 
20100812 

From Dave: Hi
I was wondering if a function can be differentiable at its endpoint. For example if I have Y = X^2 and it is bounded on closed interval [1,4], then is the derivative of the function differentiable on the closed interval [1,4] or open interval (1,4). They always say in many theorems that function is continuous on closed interval [a,b] and differentiable on open interval (a,b) and an example of this is Rolle's theorem. Thank you for your help. Answered by Robert Dawson. 





Find the speed of the current of the river 
20100519 

From paul: a boat travels 100 miles downstream in the same length of time it travels 50 miles upstream. if the boat's speed in still water is 30 mph find the speed of the current of the river? Answered by Penny Nom. 





The value of 1 acre 
20100513 

From Rebecca: If 38,121 sq feet equals $19,846.58 at 75% of its current value, what would the value of 1 acre be at 100% of value? Answered by Penny Nom. 





Sammy and his motorboat 
20100511 

From Jordan: Sammy owns a motorboat that travels 4 miles/hour in still water. It goes 12 miles upstream and 12 miles back again (downstream) in a total of 8 hours. Find the speed of the current of the river. Answered by Penny Nom. 





The rate of change of y with respect to x 
20100429 

From Tom: I just had a quick calc question about wording that wasn't ever
addressed in class. When the book says "the rate of change of y with
respect to x", should it be considered how fast y is changing in
comparison to x?
I ask because the textbook says that "y is changing 3 times faster than x,
so the rate of change of y with respect to x is 3." I'm use to rate being
like velocity, as in units of distance per units of time. All we're told
in class is that it's the slope of the tangent line, I was hoping you
could clarify for me what exactly is meant by the wording of a "rate of
change of something with respect to something else". More specifically, what
"rate" and "with respect to" mean within this context?
Thanks for your time Answered by Harley Weston. 





The derivative of cos^3x 
20100406 

From Erson: Find y' of the given function: y = cos^3x. Answered by Harley Weston. 





The nth derivative of x^(n1) log x 
20100310 

From shambodeb: This is a successive differentiation problem by Leibnitz theorem
If y = x^{n1} log x ; Proof nth derivative y^{(n)} = (n1)!/x Answered by Harley Weston. 





Painting a dome 
20091030 

From Jessica: A hemispherical dome with a radius of 50 ft will be given a coat of paint .01 inch thick.
The Contractor for the job wants to estimate the number of gallons of paint needed.
Use a differential to obtain an estimate (231 cubic inches/gallon) HINT: Approximate the change
in volume of hemisphere corresponding to increase of .01 inch in the radius. Answered by Robert Dawson. 





Differentiating y= square root(x1) 
20090929 

From edith: describe the xvalues at which f is differentiable.
y= square root(x1) Answered by Penny Nom. 





Maximum Volume of a Cylinder Inscribed in a Sphere 
20090618 

From Jim: Hello I have a hard time finishing this question:
A right circular cylinder has to be designed to sit inside a sphere of radius 6 meters
so that each top and bottom of the cylinder touches the sphere along its complete
circular edge. What are the dimensions of the cylinder of max volume and what is the volume? Answered by Janice Cotcher. 





differentiate y sin[x^2]=x sin[y^2] 
20090511 

From mamiriri: derivate y sin[x^2]=x sin[y^2] Answered by Harley Weston. 





Implicit differentiation 
20090301 

From Emily: determine the derivative y' at the point (1,0)
y= ln(x^2+y^2)
y'(1)= ?? Answered by Stephen La Rocque. 





Multiplying in different bases 
20090225 

From Susan: 11 base 2 X 22 base 3 + 33 base 4 = _________ base 5 Answered by Robert Dawson. 





Fred in his canoe 
20090221 

From Susan: Fred paddled for 4 hours with a 7 km/hr current to reach a campsite. The return trip against the same current took 9 hr. Find the speed of Fred's canoe in still water. (Remember to include correct units) Answered by Penny Nom. 





Implicit differentiation 
20090218 

From Sunny: Find slope of the tangent line to the curve 2(x^2+y^2)2=25(x^2–y^2) at (3,1) Answered by Robert Dawson and Harley Weston. 





Rent and a commission 
20090217 

From cliff: a landlord receives $95.00 rent 5% commission was paid to get to $95.00 what was the original number. The answer is obvious but what is the formula. Thanks. Cliff Answered by Penny Nom. 





In what base is 3x3= 10? 
20090214 

From David: In what base is 3x3= 10, 3x3=11, 3x3 = 12? is there a fast way to see this or do I have to create multiplication tables until I find the right one? Answered by Penny Nom. 





What's the derivative of y = e^2xe^2x / e^2x+e^2x? 
20090202 

From Heather: What's the derivative of y = e^2xe^2x / e^2x+e^2x ? Answered by Harley Weston. 





Determine y'' by implicitly differentiating twice 
20090104 

From Walter: Given x^3  3xy + y^3 = 1 , determine y'' by implicitly differentiating
twice. I cannot solve this. Would you be kind enough to perform the
mathematics and show the steps involved in obtaining the solution? Answered by Harley Weston. 





Division in different bases 
20081202 

From MICHELLE: DIVIDE 538 BY 14 IN BASE 2, 3, 4 & 5 Answered by Penny. 





Separating variables 
20081104 

From Terry: by separating variables solve the initial value problem
(x+1)y' + y = 0 y(0) = 1 Answered by Harley Weston. 





Antiderivative of 1/(x(1  x)) 
20081022 

From Matt: derivative of dx/(x(1x))
From what I've seen I should break apart the equation as such
derivative of dx/x  dx/(1x)
and then get the 2 corresponding log functions.
If that is correct why does this factoring work, if that is incorrect what is the proper way to find the derivative. Answered by Harley Weston. 





Concavity and the second derivative 
20081015 

From Christina: I'm having trouble solving for a second derivative for the following graphing question.
f(x) = (X^2+2x+4)/2x
using the quotient rule, I found:
f'(x) = (x^24)/(2x^2)
however, using the quotient rule again I can't seem to solve it (concavity):
f'''(x)=[(2x)(2x^2)(x^24)(4x)]/[(2x^2)^2]
f''(x)=[(4x^3(4x^3 16x)]/4x^4
f''(x)=16x/4x^4
f''(x)=4/x^3
and making the equation equal to zero result in 0=4 which doesn't seem to make sense... Answered by Penny Nom. 





A different approach to a word problem 
20081003 

From Kenneth: Sarah's age is 2/3 of Mary's age and 3/4 of Ruth's age. The sum of their ages is 46 years. How old is each? Answered by Penny Nom. 





What is the derivative of (2^sinx)/(logbase4(2x+1))? 
20080916 

From Jesse: What is the derivative of (2^sinx)/(logbase4(2x+1)) Answered by Harley Weston. 





Trough Filling with Water 
20080821 

From lanny: a triangular trough is 10 feet long, 6 feet across the top, and 3 feet deep. if water flows at the rate of 12 cubic inches per minute, find how fast the surface is rising when the water is 6 inches deep. Answered by Janice Cotcher. 





A series solution of y' = xy 
20080703 

From sasha: I've to find the power series solution of the differential equation: y' = xy.
I don't know how to find the recursive equation. Can you please help me. Thanks Answered by Harley Weston. 





Differentiate y= (x^x^x)^x 
20080627 

From emril: Differentiate y= (x^x^x)^x Answered by Harley Weston. 





Geometric shapes that are the same 
20080609 

From tia: what is the name used for shapes that are exactly the same? Answered by Harley Weston. 





f(x)=sin^3(3x^2) find f ' (x) 
20080421 

From Michael: f(x)=sin^3(3x^2) find f ' (x) Answered by Harley Weston. 





The chain rule 
20080410 

From joey: pls help me to Differentiate
y=(3x^24x)^8 Answered by Harley Weston. 





Differentiate 
20071228 

From taiwo: i am finding it difficult to use first principle to differentiate this question: y=xcos2x. can u help me. Answered by Penny Nom. 





The tangent to a curve 
20071210 

From Christy: I know this question is simple but I can't figure out what I'm doing wrong.
Find the equation of the tangent line to the curve 2x^2  y^4= 1 at the point (1,1). Answered by Penny Nom. 





The derivative 
20071119 

From ralf: Find the derivative of the function
1. y=1+2x^{8}
2. y=(1+2x )^{8} Answered by Harley Weston. 





f(x+y) = f(x) + f(y) + 2xy 
20071101 

From Marcia: For all real numbers x and y, let f be a function such that f(x+y) = f(x) + f(y) + 2xy and such that the limit as h > 0 of f(h) / h = 7, find: f(0), use the definition of the derivative to find f'(x), and find f(x). Answered by Penny Nom. 





The rate of change of the concentration of a solution 
20071030 

From Nicholas: A barrel initially has two kg of salt dissolved in twenty liters of water. If water flows in the rate of 0.4 liters per minute and the wellmixed salt water solution flow out at the same rate, how much salt is present after 8 minutes?
I tried working backwards given the answer but I can't seen to get their answer of ~1.7kg. Any help would be great! Thanks Answered by Harley Weston. 





What is the rent? 
20071026 

From Mark: Rent was $1.00 per cubic yard. Dimensions were 36 feet by 200 feet by 12 feet high. What is the rent Answered by Penny Nom. 





The derivative of f(x)=1/(x1) 
20070921 

From Michelle: im having trouble finding the derivative of f(x)=1/(x1) using the f(x+h)f(x)/h method. Answered by Stephen La Rocque. 





Differentiate x^(1/3) using first principles 
20070914 

From Sheila: our teacher gave us this question as a challenge and even he couldnt figure it out:
Differentiate x^(1/3) [aka the cube root of x] using first principles. i know the answer is 1/(3.x^2/3), but how is it possible using first principles? Answered by Harley Weston. 





Solve y'' + y = 0 
20070728 

From Shihya: How do you solve y’’ + y = 0 Answered by Stephen La Rocque and Harley Weston. 





Implicit Derivatives 
20070713 

From Charles: I need help computing y' by implicit differentiation the question is:
y^2 + x/y + 4x^2  3 Answered by Stephen La Rocque. 





sinx and cosx 
20070625 

From Mac: Can anyone tell me whether sinx and cosx is differentiable at x=0 ?
As far as i know, cos(x) and sin(x) is differentiable at all x. Answered by Penny Nom and Stephen La Rocque. 





Rent per square foot per month 
20070416 

From julie: my property is 60'x66'
rent is $700 a month
how much per inch per month does this equal
how much per foot per month does this equal Answered by Penny Nom. 





The second derivative 
20070414 

From Gerry: In mathematical context,what do you understand by the term "Second Derivative" Answered by Penny Nom. 





Velocity 
20070404 

From Jessica: The speed of the current in a river is 2mph. Jessica travels 20 miles upstream and then 20 miles downstream in a total time of 5 1/3 hr. Find the speed of the boat.
I know you have to use V= d/t, but what is the equation for time because you have to factor in that upstream the boat is slower and downstream the boat is faster? Answered by Penny Nom. 





What is the intensity 5m below the surface? 
20070331 

From david: I have this question which I am supposed to set it up and solve as a differential equation. I know how to solve the diffrential equation but I am having hard time understanding this question. Here is the question: The intensity of light in the ocean decreases the deeper you dive. In fact, the rate at which the intensity decreases is proportional to the current intensity. Setup the corresponding differential equation and solve for I(Y), the intensity I as a function of current intensity Y. If the light intensity 2m below the surface is 25% of the intensity at the surface, what is the intensity 5m below the surface. Can you please explain to me what does it mean by current intensity and how do I set this equation up. Thanks for the help. Answered by Penny Nom. 





The period of a simple pendulum 
20070310 

From Melissa: The period of a simple pendulum of length L feet is given by: T=2pi(sqrt(L/g))seconds. It is assumed that g, the acceleration due to gravity on the surface of the earth, is 32 feet per second per second. If the pendulum is a clock that keeps good time when L=4 feet, how much time will the clock gain in 24 hours if the length of the pendulum is decreased to 3.97 feet? (Use differentials and evaluate the necessary derivative at L=4 feet.) Answer is in seconds. Melissa Answered by Penny Nom. 





Derivatives 
20070201 

From Jacob: Find the derivative of: y= pi^2+x^2+3xy+sin(y^2) Answered by Penny Nom. 





Water is being pumped into the pool 
20061024 

From Jon: A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deeps at the deep end. Water is being pumped into the pool at 1/4 cubic meters per minute, an there is 1 meter of water at the deep end.
a) what percent of the pool is filled?
b) at what rate is the water level rising? Answered by Stephen La Rocque. 





Brackets and more brackets 
20060829 

From Michelle: Feeling stupid asking but it's been awhile ... {{{2}}} ...what is this really saying ....are the outer brackets = null? Answered by Stephen La Rocque. 





Differentiate Y= sin3x + cos7x 
20060822 

From james: Differentiate the function of x using the basic rules.
Y= sin3x + cos7x Answered by Stephen La Rocque. 





differentiate the volume of a cylinder with V respect to h 
20060524 

From A student: differentiate the volume of a cylinder with V respect to h Answered by Stephen La Rocque. 





Rate of ladder falling 
20060430 

From Harsh: A ladder 4 m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 30 cm/s, how quickly is the top of the ladder sliding down the wall when the bottom of the ladder is 2 m from the wall? Answered by Stephen La Rocque. 





Differentiation, powers and logs 
20060106 

From Claudia:
Question: how do I find the derivative of
x* ln(x+(e^2))^2
x^lnx
x^(e^(x^2))
Answered by Penny Nom. 





Notation for the second derivative 
20051108 

From Mussawar: my question is ^{d}/_{dx}( ^{dy}/_{dx}) = ^{d2y}/_{dx2}. why it is not equal to ^{d2y}/_{d2x}. Answered by Penny Nom. 





Velocity and acceleration 
20051027 

From Candace: When taking the integral of the position function, you get the velocity function, and the same for velocity to acceleration. So when you do each of these, you get a function. But when you integrate on a graph, you get an area under a curve. The area is un units squared where do the units go when you make it an equation? How can a function be an area? Answered by Harley Weston. 





Can we take the derivative of independent variable 
20051018 

From Mussawar: why we take derivative of dependent variable with respect to independent variable .can we take the derivative of independent with respect to dependent.if not why. Answered by Walter Whiteley. 





U'(X)  U(X) = 0; U(0) = 2 
20050923 

From David: Out of interest could you please answer the following questions?
U'(X)  U(X) = 0; U(0) = 2
and
U''(X)  U'(X) = 0; U'(0) = U(0) = 2
Answered by Harley Weston. 





How do you differentiate y=(x)^(x^x)? 
20050914 

From Calebius: How do you differentiate y=(x)^{(xx)}? Answered by Penny Nom. 





Logarithmic differentiation 
20050523 

From Richard: I need to convince myself that I understand the process of
differentiating y=x^{x}.
The specific question is that if I have to take the logarithm of both sides
of the equation how can differentiate the following?
y= {(x+2)^{(x+2)}}/{(x+1)^{(x+1)}}  {(x+1)^{(x+1)}}/(x^{x}),
I have an idea that the differential of this fairly complex function
is itself ... am I right or wrong. Answered by Penny Nom. 





Differentiating F(x,y) = 0 
20050123 

From Jacob: In calculus, we often mention to the students that if F(x,y) = 0, then we can differentiate both sides and still get an equality. The problem is that we can't perform the same operation on F(x) = 0, say x = 0, otherwise 1 = 0, which is absurd. What is the reason? Answered by Walyer Whileley and Harley Weston. 





ln(x)/x 
20041211 

From Tina: What is the derivative of (ln x)/x? The double derivative? Answered by Penny Nom. 





An ODE 
20041110 

From David: I have a question that i really cant do, it is as follows:
The ODE dy/dx + 0.5y = 0.5e^(1.5x) ; y(5) = 2
Solve the ODE subject to the given condition using exact methods and evaluate the solution y for x = 5 x=5.2 x=5.4 x=5.6 x=5.8 x=6 Answered by Harley Weston. 





Implicit differentiation 
20041024 

From Emily: If x^3+3xy+2y^3=17, then in terms of x and y, dy/dx = Answered by Penny Nom. 





differentiate Y=X^X^X 
20040913 

From Kunle: differentiate Y=X^X^X Answered by Penny Nom. 





The integrating factor method 
20040805 

From A student: Whilst using the integrating factor method, I am required to integrate a function multipled by another function.
say f(t) = exp(kt) and some other function g(t); where exp = exponential and k is some constant.
Integral f(t)*g(t) dt or
Integral exp(kt)*g(t) dt
What would the result of this integral be? I have never met an integral like this before. Would it simply be exp(kt)*g(t)/k?
More specifically, the problem and my attempted answer is in PDF format:
In my attempted solution, I am unsure about the last two lines I have written out, as it relates to integrating a function multipled by another function. Answered by Harley Weston. 





Differentiation 
20040804 

From A parent: I am a parent trying to understand higher level of maths and would be very grateful if you could help with differentiating the following functions, identifying general rules of calculus:
a) f(x)=e^2^xIn(cos(8x))
b) f(x)=secx/SQRTx^4+1 Answered by Penny Nom. 





Some calculus problems 
20040401 

From Weisu:
I have questions about three word problems and one
regular problem, all dealing with derivatives.
 Find all points on xy=e^{xy} where the tangent line
is horizontal.
 The width x of a rectangle is decreasing at 3 cm/s,
and its length y is increasing at 5 cm/s. At what rate
is its area A changing when x=10 and y=15?
 A car and a truck leave the same intersection, the
truck heading north at 60 mph and the car heading west
at 55 mph. At what rate is the distance between the
car and the truck changing when the car and the truck
are 30 miles and 40 miles from the intersection,
respectively?
 The production P of a company satisfies the
equation P=x^{2} + 0.1xy + y^{2}, where x and y are
the inputs. At a certain period x=10 units and y=8
units. Estimate the change in y that should be made to
set up a decrease of 0.5 in the input x so that the
production remains the same.
If you could just give me some hints on these
questions, I'd really appreciate it. Thanks! Answered by Penny Nom. 





A derivative 
20040331 

From A student: What is the nth derivative of f(x) =(2x)/(1(x^{2}))? Answered by Harley Weston. 





A partial derivative 
20040319 

From Penny Nom: Is it possible to differentiate the following equation, if so could
you please explain.
S=SQRT(T(5/X^2))
I would like the derivative of S with respect to X. Answered by Harley Weston. 





The derivative of x to the x 
20040214 

From Cher: what about the derivative of x to the power x? Answered by Penny Nom. 





The slope of a tangent 
20031001 

From A student:
find the slope of the tangent to each curve at the given point f(x)=square root 16x, where y=5 Answered by Penny Nom. 





Differentiating inverses 
20021120 

From Amy: f(x)= x^{3}+x+1, a=1 find g'(a) (g = f^{ 1}). I am having trouble finding g(a). Answered by Penny Nom. 





Monthly rent 
20020904 

From Al: How do you figure monthly rent if the charge is 11.00 per sq. ft. and there are 1,000 sq. feet? Answered by Leeanne Boehm. 





Undetermined coefficients 
20011122 

From Hoda: The equation is: y"  2y' + y = t e^{t} + 4 We need to use The method of Undetermined coefficients. I have tried assuming that the solution is Ate^{t}+Be^{t}+C, but all I get is C=4 and I tried (At^{2}+Bt+C)e^{t}+D, but again I get 0=0 when I calculate the first and second derivatives, so i get no information on the constants. Any suggestions? Answered by Harley Weston. 





A tangent line 
20011121 

From A student: write an equation of the line tangent to the graph of
e^{y} + ln(xy) = 1 + e at (e,1) Answered by Harley Weston. 





4 sinx cosy = 1 
20011010 

From A student: How would i differentiate the following example in terms of t (x and y are functions of t) 4 sinx cosy = 1 Answered by Claude Tardif. 





(x^25x6)/(x6) 
20011002 

From Bill: given f(x) = (x^{2}5x6)/(x6) find f'(6). Answered by Harley Weston. 





National consumption function 
20010509 

From Brian: If consumption is $11 billion when disposable income is 0 and the marginal propensity to consume is dC/dy = 1/(2y+4)1/2+0.3(in billions of dollars), find the national consumption function. Answered by Harley Weston. 





Differentiation 
20010417 

From Esther: Could you please tell me what the first derivative is of the following: y = 2/(2x+e^{2x}) Is it (1+xe^{2x})/(2x+e^{2x})^{2} or perhaps 4(1+e^{2x})/(2x+e^{2x})^{2} ? I am a little confused between the two! Answered by Harley Weston. 





Airflow in windpipes 
20010325 

From Ena: The volume of air flowing in windpipes is given by V=kpR^{4}, where k is a constant, p is the pressure difference at each end, R is the radius. The radius will decrease with increased pressure, according to the formula: Ro  R = cp, where Ro is the windpipe radius when p=0 & c is a positive constant. R is restricted such that: 0 < 0.5*Ro < R < Ro, find the factor by which the radius of the windpipe contracts to give maximum flow? Answered by Harley Weston. 





A jogger 
20010312 

From Bill: At time t=0 a jogger is running at a velocity of 300 meters per minute. The jogger is slowing down with a negative acceleration that is directly propotional to time t. This brings the jogger to a stop in 10 minutes. a) write an expression for the velocity of the jogger at time t. b) what is the total distance traveled by the jogger in that 10minute interval. Answered by Harley Weston. 





The domain of the derivative 
20010222 

From Wayne: I know that the domain of f'(x) is a subset of the domain of f(x). Is it necessarily true that the subset will have at most one less element than the domain of the original function? Answered by Harley Wesston. 





Differentiation of y = x^{ n} 
20010217 

From Jashan: i am studying differentation at the moment i have drawn some graphs such as y=x^{ 2}. i have found the formula for the gradient of this curve, this being 2x obtained by using differentation, but i need to know the general case for the formula where y=x^{n } in order for me to understand this topic more throughly, i would also like to know how u derived this general formula Answered by Harley Weston. 





Find an exprression for f(x) 
20010207 

From A 12th grade AP Calc student: Let f be the function defined for all x > 5 and having the following properties. Find an expression for f(x). i) f^{ ''}(x) = 1/ (x+5)^{1/3} for all x in the domain of f ii) the line tangent to the graph of f at (4,2) has an angle of inclination of 45 degress. Answered by Harley Weston. 





23 + 17 * 2 / 8 + 11 = 13 
20010202 

From David: My child is working through basic operations, and gave this problem: 23 + 17 * 2 / 8 + 11 = 13 The teacher wants the parenthesis to be put in the right places to come up with 13. Answered by Claude Tardif. 





How do you integrate secant(theta)? 
20001222 

From Robert Williamson: How do you integrate secant(theta)? I know the answer is ln [sec(theta) + tan(theta)] but how do you get there? Answered by Claude tardif. 





A point that a group of lines pass through 
20001205 

From Ross: A point that a group of lines pass through is called a________? Answered by Chris Fisher. 





Derivatives, there must be an easier way 
20000906 

From Brad Goorman: The direction read: Take the derivative of each expression. y = {1+[x+(x^{2} +x^{3})^{4}]^{5}}^{6}
Answered by Harley Weston. 





A derivative problem 
20000604 

From Jeff Ellis: If F(x)=(4+x)(3+2x^{2})^{2}(2+3x^{3})^{3}, find F'(0) Answered by Harley Weston. 





Thearcius Functionius 
20000503 

From Kevin Palmer: With the Olympics fast approaching the networks are focusing in ona new and exciting runner from Greece. Thearcius Functionius has astounded the world with his speed. He has already established new world records in the 100 meter dash and looks to improve on those times at the 2000 Summer Olympics. Thearcius Functionius stands a full 2 meters tall and the networks plan on placing a camera on the ground at some location after the finish line(in his lane) to film the history making run. The camera is set to film him from his knees(0.5 meters up from the ground) to 0.5 meters above his head at the instant he finishes the race. This is a total distance of two meters(the distance shown by the camera's lens). Answered by Harley Weston. 





An indefinite integral 
20000503 

From Bonnie Null: I am to find the indefinite integral of: (e^{x}  e^{x})^{2} dx Answered by Claude Tardif. 





Paying with silver 
20000426 

From Luther Jackson: A silver prospector is unable to pay his March rent in advance. He owned a bar of pure silver, 31 inches long, so he made the following arrangement with his landlady. He would cut the bar, he said into smaller pieces. On the first day of March he would give her and inch of the bar, and on each succeeding day he would add another inch to her amount of silver. She would keep this silver as security. At the end of the month, when the prospector expected to be able to pay his rent in full, she would return the pieces to him. . . . Answered by Claude Tardif and Penny Nom. 





y = x^x^x^x... 
20000405 

From Michael Hackman: Find the derivative of: y = x^x^x^x... on to infinity. Answered by Claude Tardif. 





A mixture problem 
20000306 

From Rebecca Edwards: A tank in which cholocate milk is being mixed contains a mixture of 460 liters of milk and 40 liters of chocolate syrup initially. Syrup and milk are then added to the tank at the rate of 2 liters per minute of syrup and 8 liters of milk per minute. Simultaneously the mixture is withdrawn at the rate of 10 liters per minute. Find the function giving the amount of syrup in the tank at time t. Answered by Harley Weston. 





A calculus problem 
19991208 

From JT Wilkins: These are the questions:  Show that there exists a unique function that meets the following requirements:
a) f is differentiable everywhere b) f(0)= f'(0)= 0 c) f(x+y)= f(x)+ f(y), for all real values of x,y  Consider the function F: R>R (All Reals)
F(x) = 0, for x irrational & 1/q, x=p/q gcd(p,q)=1 q > 0 a)determine the values x where f is continuous, respectively discontinuous. b)determine the values x when f is differentiable and for each of these values compute f'(x). Answered by Penny Nom. 





The chain rule 
19991203 

From Jennifer Stanley: This problem is making me dizzy. I would greatly appreciate a little help! Express the derivative dy/dx in terms of x. y=u^2(uu^4)^3 and u=1/x^2 Answered by Harley Weston. 





Two calculus problems 
19991201 

From O'Sullivan: Question #1 Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely? It's under the chain rule section of differentiation if that any help. I've set up a ratio and tried to find the constant but am stuck. Question #2 The figure shows a lamp located three units to the right of the yaxis and a shadow created by the elliptical region x^2 + 4y^2 < or= 5. If the point (5,0) is on the edge of the shadow, how far above the x axis is the lamp located? The picture shows an x and y axis with only the points 5 and 3 written on the x axis. the lamp is on the upper right quadrant shining down diagonally to the left. There's an ellipse around the origin creating the shadow. It's formula is given as x^2 + 4y^2=5. Answered by Harley Weston. 





Two derivatives 
19991116 

From Gina Renicker: The derivative of: y=e^{(xlnx)} and y=x^{2arctan(x1/2)} Answered by Harley Weston. 





Clockwise or Counterclockwise? 
19991027 

From Tim: A particle moves around the circle x^{2} + y^{2} = 1 with an xvelocity component dx/dt = y  Find dy/dt
 Does the particle travel clockwise or counterclockwise around the circle? Why?
Answered by Harley Weston. 





Derivatives with logs 
19991026 

From Kate: What is the derivative of 5 to the 5x2 at x equals 0.8? Answered by Harley Weston. 





Parametric Equations 
19990806 

From Nicholas Lawton: Show that an equation of the normal to the curve with parametric equations x=ct y=c/t t not equal to 0, at the point (cp, c/p) is : yc/p=xp^2cp^3 Answered by Harley Weston. 





A calculus problem 
19990722 

From Nicholas Lawton: The curve y= e^x(px^2+qx+r) is such that the tangents at x=1 and x=3 are parallel to the xaxis. the point (0,9) is on the curve. Find the values of p,q and r. Answered by Harley Weston. 





Related rates 
19990513 

From Tammy: The sides of a rectangle increase in such a way that dz/dt=1 and dx/dt=3*dy/dt. At the instant when x=4 and y=3, what is the value of dx/dt? (there is a picture of a rectangle with sides x and y, and they are connected by z, which cuts the rectangle in half) Answered by Harley Weston. 





Graphing the Derivative 
19990118 

From Milena Ghebre: This question has been nagging me for sometime now. Is there a way of finding out the derivative of a function, just by looking at the graph of it? Answered by Walter Whiteley. 





Calculus 
19990116 

From Kaylea Rankin: Differentiate the following. y = 1 /(2+3/x) Answered by Jack LeSage and Penny Nom. 





The area and the circumference of a circle. 
19980827 

From Jason Wright: I was looking at the relationship of the area of a circle and the circumference when I realized that 2*pi*r is the derivative of pi*r^2. I was wondering if there is any connective deep dark meaning as to why this appears to be related. Thanks for any help you can give me! Answered by Walter Whiteley. 





Concurrent Lines in a Triangle 
19980810 

From Chris Woolf: The question is Name four types of concurrent lines, rays, or segments that are associated with triangles. Answered by Chris Fisher. 





A Tightrope Walker. 
19980219 

From Amy Zitron: A tightrope is stretched 30 feet above the ground between the Jay and the Tee buildings, which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second from point A to point B, is illuminated by a spotlight 70 feet above point A.... Answered by Harley Weston. 





Mathematical Induction and the Derivative 
19970318 

From Shuling Chong: "Obtain a formula for the nth derivative of the product of two functions, and prove the formula by induction on n." Any educated tries are appreciated. Answered by Penny Nom. 





Une autre bille de rayon différent 
20020227 

From Sarah: une bille de 6 cm repose au fond d'1 cylindre droit dont la base est un disque de rayon 10 cm. On verse de l'eau dans le cylindre de façon à recouvrir exactement la bille. Démontrez que l'on peut remplacer la bille par une autre bille de rayon différent (mais supérieur) de sorte que l'eau initialement versée recouvre exactement cette nouvelle bille Answered by Claude Tardif. 





les fractions 
19980224 

From Colette Huguenin: Bonjour je révise mes math de tout les secondaire et le livre louer la bibliothèque n'est pas entier voici le genre de problêmes qui me bloque (5 4/5+1/2)divisé 1/3 je sais que je dois commencer par la parenthèse mais...je fait quoi comme opération avec le 5? doisje le multiplier ou l'additionner ou.......????? si je pouvais seulement avoir la base des fractions je redébloquerais surement Answered by Diane Hanson. 

