Question

The derivative of $\cot^{-1}x$ with respect to $\log(1+x^2)$ is

• A. $-2x$
• B. $-\dfrac{1}{2x}$
• C. $\dfrac{1}{2x}$
• D. $2x$

Hint:

For derivatives with respect to another function, use the formula $\dfrac{dy}{du} = \dfrac{dy/dx}{du/dx}$.

Answer 1

The Correct Option is B

Step 1: Writing the required derivative.
\[ \frac{d(\cot^{-1}x)}{d(\log(1+x^2))} = \frac{\frac{d}{dx}(\cot^{-1}x)}{\frac{d}{dx}(\log(1+x^2))} \]
Step 2: Differentiating numerator and denominator.
\[ \frac{d}{dx}(\cot^{-1}x) = -\frac{1}{1+x^2} \] \[ \frac{d}{dx}(\log(1+x^2)) = \frac{2x}{1+x^2} \]
Step 3: Simplifying.
\[ \frac{-\frac{1}{1+x^2}}{\frac{2x}{1+x^2}} = -\frac{1}{2x} \]
Step 4: Conclusion.
The required derivative is $-\dfrac{1}{2x}$.