Let f(x)and g(x)be two functions.Values of f(x), f '(x), g(x), and g'(x)for x = 0, 1, and 2 are given in the table below.Use the information in the table to find $H ^ { \prime } ( 0 )$ if $H ( x ) =$ e ^g⁽^x⁾ $$+ \pi$$ x. $\begin{array} { c c c c c } \boldsymbol { x } & f ( x ) & f ^ { \prime } ( x ) & g ( x ) & g ^ { \prime } ( x ) \ 0 & 1 & - 1 & 2 & 5 \ 1 & - 1 & 2 & 4 & 0 \ 2 & 7 & 3 & 11 & 0.5 \end{array}$ A) $5 e ^ { 2 }$ B) $5 e + \pi$ C) $e ^ { 2 } + \pi$ D) $5 e ^ { 2 } + \pi$

Question

Quizplus · Accepted Answer

To find $H'(0)$, we differentiate $H(x) = e^{g(x)} + \pi x$. Using the chain rule and the derivative of a linear function, $H'(x) = e^{g(x)} \cdot g'(x) + \pi$. Substituting $x = 0$, $g(0) = 2$, and $g'(0) = 5$, we get $H'(0) = e^{2} \cdot 5 + \pi$, which matches choice D.

Let F(x)and G(x)be Two Functions H′(0)H ^ { \prime } ( 0 )H′(0)

Let F(x)and G(x)be Two Functions $H ^ { \prime } ( 0 )$