Question

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

Answer 1

Find derivative of $\sin \left(2x\right)$

Let $y=\sin \left(2x\right)$

Differentiate it w.r.t $x$ using chain rule.

$$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}\sin \left(2x\right) \\ =\cos \left(2x\right)\frac{d}{dx}\left(2x\right)\left[\mathrm{chain}\mathrm{rule}:\frac{d}{dx}f\left(g\left(x\right)\right)=f\text{'}\left(g\left(x\right)\right)\frac{d}{dx}g\left(x\right)\right] \\ =\cos \left(2x\right)\times 2 \\ =2\cos \left(2x\right) \end{gathered}$$

Hence, the derivative of $\sin \left(2x\right)$ is $2\cos \left(2x\right)$.