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Question

Mathematics Question on Derivatives

If y=cot⁡−1(tan⁡x2),y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right), then dydx\frac{dy}{dx} is equal to

A

12\frac{1}{2}

B

00

C

x2\frac{x}{2}

D

−12-\frac{1}{2}

Answer

−12-\frac{1}{2}

Explanation

Solution

y=cot⁡−1(tan⁡x2)y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right) y=cot⁡−1cot⁡(π2−x2)y={{\cot }^{-1}}\cot \left( \frac{\pi }{2}-\frac{x}{2} \right) y=π2−x2y=\frac{\pi }{2}-\frac{x}{2}
Differentiating w.r.t. xx on both sides,
dydx=−12\frac{dy}{dx}=-\frac{1}{2}