# 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, $e^{x}$ :
$(e^{x})'\,=\,e^{x}.$ Solution:

Step 1:
We need to start by identifying the two functions that are being multiplied together so we can apply the product rule.
$g(x)\,=\,2x^{3},\,$ and $\,h(x)\,=\,e^{3x+5}$ Step 2:
We can now apply the three advanced techniques.This allows us to see that

${\begin{array}{rcl}f'(x)&=&2(x^{3})'e^{3x+5}+2x^{3}(e^{3x+5})'\\&=&6x^{2}e^{3x+5}+2x^{3}(3e^{3x+5})\\&=&6x^{2}e^{3x+5}+6x^{3}e^{3x+5}\end{array}}$ $6x^{2}e^{3x+5}+6x^{3}e^{3x+5}$ 