Chapter 6.3, Problem 4QR

To determine

To find: $\frac{dy}{dx}$

Answer to Problem 4QR

  $\frac{dy}{dx}=\cot x$

Explanation of Solution

Given information: $y=\ln (\sin x)$

Calculation:

  $y=\ln (\sin x)$

Differentiating both sides of the equation with respect to x ,

  $\begin{array}{l}\frac{d}{dx}(y)=\frac{d}{dx}(\ln (\sin x))\\ \Rightarrow \frac{dy}{dx}=\frac{1}{\sin x}\frac{d}{dx}(\sin x)[\because \frac{d}{dx}(\ln x)=\frac{1}{x}]\\ \Rightarrow \frac{dy}{dx}=\frac{1}{\sin x}(\cos x)[\because \frac{d}{dx}(\sin x)=\cos x]\\ \Rightarrow \frac{dy}{dx}=\cot x\end{array}$

Thus, $\frac{dy}{dx}=\cot x$