Chapter 4.1, Problem 17E

To determine

To calculate:The differentiation of the given function.

Answer to Problem 17E

The differentiation is ${\sin }^{2}x(4\sin x{\sec }^{2}4x+3\cos x)$ .

Explanation of Solution

Given information : Given function is $y={\sin }^{3}x\tan 4x$ .

Formula used: $\frac{d(x^n)}{dx}=n{x}^{n-1},\frac{d(\sin x)}{dx}=\cos xand\frac{d(\tan x)}{dx}={\sec }^{2}x$ with product and chain rule.

Calculation:

  $y={\sin }^{3}x\tan 4x$

Differentiate with respect to x :

  $\begin{array}{l}\frac{dy}{dx}={\sin }^{3}x\times \frac{d(\tan 4x)}{d(4x)}\times \frac{d(4x)}{dx}+\frac{d({\sin }^{3}x)}{d(\sin x)}\times \frac{d(\sin x)}{dx}\\ \frac{dy}{dx}={\sin }^{3}x\times ({\sec }^{2}4x)\times 4+3\times \sin x\times \cos x\\ \frac{dy}{dx}={\sin }^{2}x\times (4\times \sin x\times {\sec }^{2}4x+3\times \cos x)\end{array}$