# Differentiating with respect to x^2 and 1/(1-x)... AP review problem

• Dec 29th 2012, 04:02 PM
zachd77
Differentiating with respect to x^2 and 1/(1-x)... AP review problem
I'm reviewing my first semester of AP Calculus. While doing some review questions, I came across these two problems.

1. If Attachment 26404, find the derivative of Attachment 26405 with respect to Attachment 26406.
2. If Attachment 26407, find the derivative of Attachment 26408 with respect to Attachment 26409.

I started manipulating the derivative notation in order to get the solution, but how do I continue this problem?
• Dec 29th 2012, 04:57 PM
skeeter
Re: Differentiating with respect to x^2 and 1/(1-x)... AP review problem
(1) $\displaystyle y = \sqrt{x^2+1}$

$\displaystyle y^2 = x^2 + 1$

$\displaystyle \frac{d}{dx^2}(y^2 = x^2 + 1)$

$\displaystyle \frac{dy^2}{dx^2}= 1$

(2) let $\displaystyle u = \frac{1}{1-x}$

$\displaystyle \frac{du}{dx} = \frac{1}{(1-x)^2}$

$\displaystyle y = x^2+x$

$\displaystyle \frac{dy}{dx} = 2x+1$

$\displaystyle \frac{dy}{du} = \frac{\frac{dy}{dx}}{\frac{du}{dx}} = (2x+1) \cdot (1-x)^2$