Question

What is the derivative of ${\sin }^2\left(x\right)$?

Answer 1

Find the derivative of ${\sin }^2\left(x\right)$.

Let, $y={\sin }^2\left(x\right)$

Differentiate both sides w.r.t $x$using chain rule .

$$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}{\sin }^2\left(x\right) \\ =2\sin \left(x\right)\times \frac{d}{dx}\sin \left(x\right)\left[\because \frac{d}{dx}{x}^n=n{x}^{n-1}\right] \\ =2\sin \left(x\right)\times \cos \left(x\right)\left[\because \frac{d}{dx}\sin \left(x\right)=\cos \left(x\right)\right] \\ =2\sin \left(x\right)\cos \left(x\right) \\ =\sin 2x\left[\because 2\sin \left(x\right)\cos \left(x\right)=\sin 2\left(x\right)\right] \end{gathered}$$

Therefore, the derivative of ${\sin }^2\left(x\right)$ is $\sin 2x$.