Hi please can you help?
Your name. Richard
Who you are, a student, a teacher, a parent, other? A teacher doing personal research
The level of the question :secondary (10-12)+

Question:

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.

Hi Richard,

I would let

y₁ = {(x+2)^(x+2)}/{(x+1)^(x+1)}

and

y₂ = {(x+1)^(x+1)}/(x^x)

and differentiate each separately and then

y' = y₁' - y₂'

To differentiate y₁ first take the natural log of both sides

ln( y₁) = ln( (x+2)^(x+2)) - ln((x+1)^(x+1)) = (x+2)ln(x+2) - (x+1)ln(x+1)

Now differentiate both sides

 ^y₁'/_y1 = (1 + ln(x+2)) - (1 + ln(x+1)) = ln(x+2) - ln(x+1) = ln[(x+2)/(x+1)]

so

y₁' = y₁{ln(x+2) - ln(x+1)} = {(x+2)^(x+2)}/{(x+1)^(x+1)}ln[(x+2)/(x+1)]

Now find y₂' and y' = y₁' - y₂'.

As you will see y and y' are different.

Penny

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