Chapter 4.1, Problem 73E

To determine

To find: $\frac{dy}{dx}$ of $y={\cos }^2\left(x^3+x^2\right)$ .

Answer to Problem 73E

Option C is the right answer because $\frac{dy}{dx}$ of $y={\cos }^2\left(x^3+x^2\right)$ is,

  $\frac{dy}{dx}=$ $-2\left(3x^2+2x\right)\cos \left(x^3+x^2\right)\sin \left(x^3+x^2\right).$

Explanation of Solution

Given information:

  $y={\cos }^2\left(x^3+x^2\right)$

calculation:

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}{\cos }^2\left(x^3+x^2\right),\text{by using the chain rule}\\ \frac{dy}{dx}=\frac{d\left[{\cos }^2\left(x^3+x^2\right)\right]}{d\left[\cos \left(x^3+x^2\right)\right]}\cdot \frac{d\left[\cos \left(x^3+x^2\right)\right]}{d\left[x^3+x^2\right]}\cdot \frac{d\left(x^3+x^2\right)}{dx}\\ \frac{dy}{dx}=2\cos \left(x^3+x^2\right)\cdot \left[-\sin \left(x^3+x^2\right)\right]\cdot \left(3x^2+2x\right)\\ \frac{dy}{dx}=-2\left(3x^2+2x\right)\cos \left(x^3+x^2\right)\sin \left(x^3+x^2\right)\end{array}$

Thus we can conclude that option C is the right answer.