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JEE Main 20246 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
6 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Differentiation

JEE Main 2024 · 6 April · Evening Shift · Mathematics · Differentiation

Suppose for a differentiable function h, h(0)=0, h(1)=1 and h^...

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

Options

  1. A5
  2. B4
  3. C8
  4. D3

Correct answer

B. 4

Step-by-step solution

g(x)=h(ex)⋅eh(x)g′(x)=h(ex)⋅eh(x)⋅h′(x)+eh(x)h′(ex)⋅exg′(0)=h(1)eh(0)h′(0)+eh(0)h′(1)=2+2=4\begin{aligned} & 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\end{aligned}
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