Question:

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

KEAM

Updated On: Jun 7, 2024

$ \frac{1}{2} $
$ 0 $
$ \frac{x}{2} $
$ -\frac{1}{2} $

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The Correct Option is D

Solution and Explanation

$ y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right) $ $ y={{\cot }^{-1}}\cot \left( \frac{\pi }{2}-\frac{x}{2} \right) $ $ y=\frac{\pi }{2}-\frac{x}{2} $
Differentiating w.r.t. $ x $ on both sides,
$ \frac{dy}{dx}=-\frac{1}{2} $

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Top Questions on Derivatives

List-I (Function)	List-II (Derivative w.r.t. x)
(A) \( \frac{5^x}{\ln 5} \)	(I) \(5^x (\ln 5)^2\)
(B) \(\ln 5\)	(II) \(5^x \ln 5\)
(C) \(5^x \ln 5\)	(III) \(5^x\)
(D) \(5^x\)	(IV) 0

Match List-I with List-II:

List-I	List-II
The derivative of $ \log_e x $ with respect to $ \frac{1}{x} $ at $ x = 5 $ is	(I) -5
If $ x^3 + x^2y + xy^2 - 21x = 0 $, then $ \frac{dy}{dx} $ at $ (1, 1) $ is	(II) -6
If $ f(x) = x^3 \log_e \frac{1}{x} $, then $ f'(1) + f''(1) $ is	(III) 5
If $ y = f(x^2) $ and $ f'(x) = e^{\sqrt{x}} $, then $ \frac{dy}{dx} $ at $ x = 0 $ is	(IV) 0

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List-I	List-II
The derivative of \( \log_e x \) with respect to \( \frac{1}{x} \) at \( x = 5 \) is	(I) -5
If \( x^3 + x^2y + xy^2 - 21x = 0 \), then \( \frac{dy}{dx} \) at \( (1, 1) \) is	(II) -6
If \( f(x) = x^3 \log_e \frac{1}{x} \), then \( f'(1) + f''(1) \) is	(III) 5
If \( y = f(x^2) \) and \( f'(x) = e^{\sqrt{x}} \), then \( \frac{dy}{dx} \) at \( x = 0 \) is	(IV) 0