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:

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

Answer 1

The Correct Option is D

To solve this problem, we must differentiate each of the functions in List I and match them with the corresponding derivatives provided in List II.

1. Derivative \(A\): \( \frac{d}{dx}[tan^{-1}(\frac{3x-x^3}{1-3x^2})] \)
   • The expression inside \(tan^{-1}\) is of the form \( \frac{u}{v} \) where \( u = 3x-x^3 \) and \( v = 1-3x^2 \).
   • Using the derivative rule for \(tan^{-1}(x)\), which is \(\frac{1}{1+x^2}\), and chain rule, the derivative is \(\frac{1}{1+\left(\frac{3x-x^3}{1-3x^2}\right)^2} \times \frac{d}{dx}(\frac{3x-x^3}{1-3x^2})\).
   • Upon differentiating and simplifying, we find it matches option \(I\): \(\frac{3}{1+x^2}\).
2. Derivative \(B\): \( \frac{d}{dx}[cos^{-1}(\frac{1-x^2}{1+x^2})] \)
   • The given function resembles a standard transformation for trigonometric functions.
   • The derivative of \(cos^{-1}(x)\) is \(-\frac{1}{\sqrt{1-x^2}}\), but here simplify the expression using trigonometric identities, leading us to \(-\frac{2}{1+x^2}\).
   • This matches with option \(IV\).
3. Derivative \(C\): \( \frac{d}{dx}[cos^{-1}(\frac{2x}{1+x^2})] \)
   • The derivative follows a similar process as \(B\), when considering inverse trigonometric transformations.
   • After applying the chain rule, we find \(-\frac{2}{1+x^2}\).
   • Thus it matches option \(III\).
4. Derivative \(D\): \( \frac{d}{dx}[cot^{-1}(\frac{3x-x^3}{1-3x^2})] \)
   • The expression is similar to \(A\), but differentiated with respect to \(cot^{-1}(x)\), which gives us \(-\frac{1}{1+x^2}.\)
   • Simplifying confirms that the derivative is \(-\frac{3}{1+x^2}\).
   • Thus it corresponds to option \(II\).

The correct matching is therefore: A-I, B-IV, C-III, D-II.