Question

The derivative of $\ln x$ with respect to $\cot x$ is

• A. $\frac{-{\sin }^{2}x}{x}$
• B. $\frac{{\cos }^{2}x}{x}$
• C. $\frac{-{\sin }^{3}x}{x}$
• D. $\frac{-{\sin }^{2}x}{x^2}$

Answer 1

The correct option is A $\frac{-{\sin }^{2}x}{x}$

Let $y=\ln x$ and $z=\cot x$
Now,
$\frac{dy}{dx}=\frac{1}{x}\frac{dz}{dx}=-{\text{cosec}}^{2}x\Rightarrow \frac{dy}{dz}=\frac{\frac{dy}{dx}}{\frac{dz}{dx}}=\frac{-1}{x\cdot {\text{cosec}}^{2}x}\therefore \frac{dy}{dz}=\frac{-{\sin }^{2}x}{x}$