Chapter 27, Problem 1RE

To determine

Whether the statement “For $y={\sin }^{3}2x,\frac{dy}{dx}=3{\sin }^{2}2x\cos 2x$” is true or false.

Answer to Problem 1RE

The given statement is $\underset{\_}{\text{false}}$.

Explanation of Solution

Formula used:

The derivative of $\sin u$ is $\frac{d\left(\sin u\right)}{dx}=\cos u\frac{du}{dx}$.

Calculation:

Evaluate the derivative of $y={\sin }^{3}2x$ as follows.

$\begin{array}{l}y={\sin }^{3}2x\\ \frac{dy}{dx}=3\left({\sin }^{2}2x\right)\left(\cos 2x\right)\left(2\right)\\ \frac{dy}{dx}=6{\sin }^{2}2x\cos 2x\end{array}$

But, it is given as $3{\sin }^{2}2x\cos 2x$.

The extra factor 2 is missing while differentiating.

Therefore, the given statement is $\underset{\_}{\text{false}}$.