Match List I with List II LIST I LIST II A . $\frac{d}{dx} [tan^{-1} (\frac{3x-x^3}{1-3x^2})]$ I . $\frac{3}{1+x^2}$ B . $\frac{d}{dx}[cos^{-1}(\frac{1-x^2}{1+x^2})]$ II . $\frac{-3}{1+x^2}$ C . $\frac{d}{dx}[cos^{-1} (\frac{2x}{1+x^2})]$ III . $\frac{-2}{1+x^2}$ D . $\frac{d}{dx}[cot^{-1}(\frac{3x-x^3}{1-3x^2})]$ IV . $\frac{2}{1+x^2}$ Choose the correct answer from the options given below:

Question

Match List I with List II LIST I   LIST II A . $\frac{d}{dx} [tan^{-1} (\frac{3x-x^3}{1-3x^2})]$ I . $\frac{3}{1+x^2}$ B . $\frac{d}{dx}[cos^{-1}(\frac{1-x^2}{1+x^2})]$ II . $\frac{-3}{1+x^2}$ C . $\frac{d}{dx}[cos^{-1} (\frac{2x}{1+x^2})]$ III . $\frac{-2}{1+x^2}$ D . $\frac{d}{dx}[cot^{-1}(\frac{3x-x^3}{1-3x^2})]$ IV . $\frac{2}{1+x^2}$ Choose the correct answer from the options given below:

cdquestions Admin · Accepted Answer

The correct option is(D):A-I, B-IV, C-III, D-II

Answer

A-II, B-III, C-I, D-IV

Answer

A-IV, B-II, C-I, D-III

Answer

A-III, B-I, C-IV, D-II

Answer

A-I, B-IV, C-III, D-II

	LIST I		LIST II
A.	\(\frac{d}{dx} [tan^{-1} (\frac{3x-x^3}{1-3x^2})]\)	I.	\(\frac{3}{1+x^2}\)
B.	\(\frac{d}{dx}[cos^{-1}(\frac{1-x^2}{1+x^2})]\)	II.	\(\frac{-3}{1+x^2}\)
C.	\(\frac{d}{dx}[cos^{-1} (\frac{2x}{1+x^2})]\)	III.	\(\frac{-2}{1+x^2}\)
D.	\(\frac{d}{dx}[cot^{-1}(\frac{3x-x^3}{1-3x^2})]\)	IV.	\(\frac{2}{1+x^2}\)

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

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

The Correct Option is D

Solution and Explanation

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