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

<p>To determine $g^{\prime}(0)$, we start by applying the chain rule and product rule to find the derivative of the given function $g(x) = h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}$.</p> <p>The product rule states that if we have two functions $u(x)$ and $v(x)$, then the derivative of their product is given by:</p> <p>$(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)$</p> <p>Let's denote $u(x) = h(\mathrm{e}^x)$ and $v(x) = \mathrm{e}^{h(x)}$.</p> <p>First, we need to find $u'(x)$ and $v'(x)$. Using the chain rule, we find:</p> <p>$$ u'(x) = \frac{d}{dx}h(\mathrm{e}^x) = h'(\mathrm{e}^x) \cdot \frac{d}{dx}(\mathrm{e}^x) = h'(\mathrm{e}^x) \cdot \mathrm{e}^x $$</p> <p>Now, the derivative of $v(x)$ is:</p> <p>$v'(x) = \frac{d}{dx}(\mathrm{e}^{h(x)}) = \mathrm{e}^{h(x)} \cdot h'(x)$</p> <p>Using the product rule, we get the derivative of $g(x)$:</p> <p>$g'(x) = u'(x) v(x) + u(x) v'(x)$</p> <p>Substituting $u(x)$, $v(x)$, $u'(x)$, and $v'(x)$ into the above expression, we get:</p> <p>$$ g'(x) = \left( h'(\mathrm{e}^x) \mathrm{e}^x \right) \mathrm{e}^{h(x)} + \left( h(\mathrm{e}^x) \right) \left( \mathrm{e}^{h(x)} h'(x) \right) $$</p> <p>Next, we need to evaluate this at $x = 0$:</p> <p>First, we know that:</p> <p>$h(0) = 0$</p> <p>$h(1) = 1$</p> <p>$h'(0) = 2$</p> <p>$h'(1) = 2$</p> <p>Substituting $x = 0$ into the expressions, we get:</p> <p>$u(0) = h(\mathrm{e}^0) = h(1) = 1$</p> <p>$v(0) = \mathrm{e}^{h(0)} = \mathrm{e}^0 = 1$</p> <p>$u'(0) = h'(\mathrm{e}^0) \mathrm{e}^0 = h'(1) \cdot 1 = 2$</p> <p>$v'(0) = \mathrm{e}^{h(0)} h'(0) = \mathrm{e}^0 \cdot 2 = 2$</p> <p>Therefore, evaluating $g'(0)$:</p> <p>$g'(0) = \left( u'(0) v(0) \right) + \left( u(0) v'(0) \right)$</p> <p>$g'(0) = \left( 2 \cdot 1 \right) + \left( 1 \cdot 2 \right)$</p> <p>$g'(0) = 2 + 2 = 4$</p> <p>Thus, the value of $g^{\prime}(0)$ is 4, which corresponds to Option A.</p> <p>The correct answer is <strong>Option A: 4</strong>.</p>