Question

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

• A. 5
• B. 3
• C. 8
• D. 4

Answer 1

The Correct Option is D

Differentiating with respect to \( x \):

\[ g(x) = h(e^x) \times e^{h(x)} \]

\[ g'(x) = h'(e^x) \times e^{h(x)} \times h'(x) + e^{h(x)} \times h'(e^x) \times e^x \]

\[ g'(0) = h(1)e^{h(0)}h'(0) + e^{h(0)}h'(1) \]

\[ = 2 + 2 = 4 \]