Suppose for a differentiable function h, h(0)=0 , h(1)=1 and h prime (0)=h prime (1)=2 . If g(x)=h (e x ) e h(x) , then g prime (0) is equal to:

Mathematics · Methods of Differentiation

JEE Main 2024 — 6 April, Shift 2 — Question 13

Suppose for a differentiable function $h, h(0)=0$ , $h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$ . If $g(x)=h\left(e^{x}\right) e^{h(x)}$ , then $\mathrm{g}^{\prime}(0)$ is equal to:

Option A:
5
Option B:
3
Option C:
8
Option D:
4
Correct

Answer: D

Step-by-step solution

$g(x)=h\left(e^{x}\right) \cdot e^{h(x)}$

$g^{\prime}(x)=h\left(e^{x}\right) \cdot e^{h(x)} \cdot h^{\prime}(x)+e^{h(x)} h^{\prime}\left(e^{x}\right) \cdot e^{x}$

$g^{\prime}(0)=h(1) e^{h(0)} h^{\prime}(0)+e^{h(0)} h^{\prime}(1)$ $=2+2=4$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 6 April, Shift 2
Subject: Mathematics
Chapter: Methods of Differentiation
Topic: Methods of Differentiation