Single Variable Calculus, Chapter 7, Review Exercises, Section Review Exercises, Problem 60

Find $f'(x)$ suppose that $f(x) = xe^{\sin x}$. Graph of $f$ and $f'$ on the same screen and comment.


$
\begin{equation}
\begin{aligned}

\text{If } f(x) =& xe^{\sin x}, \text{ then by using Product Rule,}
\\
\\
f'(x) =& x \cdot e^{\sin x} (\cos x) + (1) e^{\sin x}
\\
\\
f'(x) =& e^{\sin x} [ x \cos x + 1]

\end{aligned}
\end{equation}
$








Based from the graph, $f(x)$ is increasing whenever $f'(x)$ is positive. On the other hand, $f(x)$ is decreasing when $f'(x)$ is negative. Thus, we can say that our answer is reasonable.