6 Attachment(s)

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.

- If Attachment 26404, find the derivative of Attachment 26405 with respect to Attachment 26406.
- 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?

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 $\displaystyle u = \frac{1}{1-x}$

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

$\displaystyle y = x^2+x$

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

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