Question

Find $\frac{dy}{dx}$, If $y=\log \left(\sin \left(3x\right)\right)$.

Answer 1

Compute the required derivative:

Given: $y=\log \left(\sin \left(3x\right)\right)$.

Differentiating with respect to $x$, we get,

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{1}{\sin \left(3x\right)}\times 3\cos \left(3x\right)\left[\because \frac{d}{dx}\log \left(f\left(x\right)\right)=\frac{1}{f\left(x\right)}\frac{d}{dx}f\left(x\right)\right] \\ \Rightarrow \frac{dy}{dx}=\frac{3\cos \left(3x\right)}{\sin \left(3x\right)} \\ \Rightarrow \frac{dy}{dx}=3\cot \left(3x\right) \end{gathered}$$

Hence , $\frac{dy}{dx}=3\cot \left(3x\right)$ is the required answer.