Match List-I with List-II (x > 0 wherever needed)
| List-I | List-II | ||
|---|---|---|---|
| Derivative of | Value | ||
| (A) | \(e^{3x}\) with respect to \(e^{x}\) | (I) | \(-x\) |
| (B) | \(x^{3}\) with respect to \(x^{2}\) | (II) | \(2\sqrt{x}\,e^{x}\) |
| (C) | \(e^{x}\) with respect to \(\sqrt{x}\) | (III) | \(3e^{2x}\) |
| (D) | \(\log_e x\) with respect to \(\displaystyle\frac{1}{x}\) | (IV) | \(\displaystyle\frac{3x}{2}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
The derivative of u with respect to v is \(\frac{du}{dv} = \frac{du/dx}{dv/dx}\).
(A) \(e^{3x}\) with respect to \(e^{x}\)
This matches (III) \(3e^{2x}\):
\( \displaystyle \frac{3e^{3x}}{e^{x}} = 3e^{2x} \)
(B) \(x^{3}\) with respect to \(x^{2}\)
This matches (IV) \(\frac{3x}{2}\):
\( \displaystyle \frac{3x^{2}}{2x} = \frac{3x}{2} \)
(C) \(e^{x}\) with respect to \(\sqrt{x}\)
This matches (II) \(2\sqrt{x}\,e^{x}\):
\( \displaystyle \frac{e^{x}}{\frac{1}{2\sqrt{x}}} = 2\sqrt{x}\,e^{x} \)
(D) \(\log_e x\) with respect to \(\frac{1}{x}\)
This matches (I) \(-x\):
\( \displaystyle \frac{\frac{1}{x}}{-\frac{1}{x^{2}}} = -x \)