022 Exam 2 Sample B, Problem 3

Find the derivative of $f(x)\,=\,2x^{3}e^{3x+5}$.

Foundations:
This problem requires several advanced rules of differentiation. In particular, you need
The Chain Rule: If $f$ and $g$ are differentiable functions, then
$(f\circ g)'(x)=f'(g(x))\cdot g'(x).$
The Product Rule: If $f$ and $g$ are differentiable functions, then
$(fg)'(x)=f'(x)\cdot g(x)+f(x)\cdot g'(x).$
Additionally, we will need our power rule for differentiation:
$\left(x^{n}\right)'\,=\,nx^{n-1},$ for $n\neq 0$,
as well as the derivative of the exponential function, $e^{x}$:
$\left(e^{f(x)}\right)'\,=\,\left(f(x)\right)'\cdot e^{f(x)}.$

 Solution:

Step 1:
We need to identify the composed functions in order to apply the chain rule. Note that if we set $g(x)\,=\,\ln x$, and
$f(x)\,=\,{\frac {(x+5)(x-1)}{x}},$
we then have  $y\,=\,g\circ f(x)\,=\,g\left(f(x)\right).$