Question

Match List I with List II

List I (Functions)		List II (Derivatives)
A.	f(x)=sin-1x	I.	\(\frac{1}{1+x^2}\), x ∈ R
B.	f(x)=tan-1x	II.	\(\frac{1}{\sqrt{1-x^2}}\), x ∈ (-1, 1)
C.	f(x)=cos-1x	III.	\(-\frac{1}{\sqrt{1-x^2}}\), x ∈ (-1, 1)
D.	f(x)=sin-1x	IV.	\(-\frac{1}{1+x^2}\), x ∈ R

Choose the correct answer from the options given below :

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

Answer 1

The Correct Option is C

To match the functions in List I with their corresponding derivatives in List II, we need to use the derivatives of inverse trigonometric functions:

1. The derivative of \( f(x) = \sin^{-1}x \) is \( f'(x) = \frac{1}{\sqrt{1-x^2}} \), which corresponds to item II.
2. The derivative of \( f(x) = \tan^{-1}x \) is \( f'(x) = \frac{1}{1+x^2} \), which corresponds to item I.
3. The derivative of \( f(x) = \cos^{-1}x \) is \( f'(x) = -\frac{1}{\sqrt{1-x^2}} \), which corresponds to item III.
4. The question includes \( f(x) = \sin^{-1}x \) again, but since functions need to be unique in the list, it likely refers to comparing repeated options. However, we compare this with \( -\frac{1}{1+x^2} \), which corresponds to IV.

Considering this reasoning, the correct matches are:

• A matches with II.
• B matches with I.
• C matches with III.
• D matches with IV.

The correct option is:

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