Question

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

Answer 1

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

Consider the function $2x$as $u$ and $\sin \left(u\right)$ as $v$.

Differentiate $\sin \left(2x\right)$ with respect to $x$ using chain rule:

$$\begin{gathered} \frac{d\left[\sin \left(2x\right)\right]}{dx}=\frac{d\left[\sin \left(u\right)\right]}{du}·\frac{du}{dx} \\ =\cos \left(u\right)·\frac{d\left(2x\right)}{dx} \\ =\cos \left(u\right)·2 \\ =2\cos \left(u\right) \end{gathered}$$

Substitute $2x$ for $u$:

$\frac{d\left[\sin \left(2x\right)\right]}{dx}=2\cos \left(2x\right)$

Hence, the derivative of $\mathbf{\text{sin(2x)}}$ is $\mathbf{\text{2cos(2x)}}$.