Match List-I with List-II (each function is taken where it is defined)
| List-I | List-II | ||
|---|---|---|---|
| Function | Derivative | ||
| (A) | \(e^{\sin x}\) | (I) | \(-\tan x\) |
| (B) | \(\log_e(\sin x)\) | (II) | \(e^{x}\cos(e^{x})\) |
| (C) | \(\sin(e^{x})\) | (III) | \(\cot x\) |
| (D) | \(\log_e(\cos x)\) | (IV) | \(\cos x\,e^{\sin x}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
Each function is an outer function of an inner one, so the chain rule applies: derivative of the outer, times derivative of the inner.
(A) \(e^{\sin x}\)
This matches (IV) \(\cos x\,e^{\sin x}\).
\( \displaystyle \frac{d}{dx}e^{\sin x} = e^{\sin x} \cdot \cos x \)
(B) \(\log_e(\sin x)\)
This matches (III) \(\cot x\).
\( \displaystyle \frac{d}{dx}\log_e(\sin x) = \frac{1}{\sin x} \cdot \cos x = \cot x \)
(C) \(\sin(e^{x})\)
This matches (II) \(e^{x}\cos(e^{x})\).
\( \displaystyle \frac{d}{dx}\sin(e^{x}) = \cos(e^{x}) \cdot e^{x} \)
(D) \(\log_e(\cos x)\)
This matches (I) \(-\tan x\).
\( \displaystyle \frac{d}{dx}\log_e(\cos x) = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x \)