   SEARCH HOME Math Central Quandaries & Queries  Question from ralf, a student: Find the derivative of the function 1-. y=1+2x8 2-. y=(1+2x )8 Ralf,

I like this problem. I'm going to do one like it for you, but with three parts.

Find the derivative of the function
1 -. y = 3 - 5x4
2 -. y = 3 - (5x)4
3 -. y = (3 - 5x )4

1. This is an application of the first properties of differentiation. the derivative of a constant is zero, the derivative of xr is rxr-1 and if k is a constant and f(x) is a differentiable function then the derivative of k f(x) is k f '(x). Thus the derivative of y = 3 - 5x4 is dy/dx = -5 (4x3) = -20 x3.

2. For the second problem you need the properties I stated before plus the chain rule. One special case of the chain rule is that if f(x) is a differentiable function and k is a constant then the derivative of (f(x))k is k (f(x))k-1 f '(x). Hence the derivative of y = 3 - (5x)4 is dy/dx = -4 (5x)3 × 5 = -20 (5x)3.

3. This is an application of the chain rule again. Since the derivative of 3 - 5x is -5 the derivative of y = (3 - 5x )4 is dy/dx = 4(3 - 5x)3 × (-5) = -20 (3 - 5x)3

I hope this helps,
Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.