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. Let's call $g(x)\,=\,2x^{3},\,$ and $\,h(x)\,=\,e^{3x+5}$, so $f(x)=g(x)\cdot h(x)$.

Step 2:
We can now apply the advanced techniques.This allows us to see that
${\begin{array}{rcl}f'(x)&=&2(x^{3})'e^{3x+5}+2x^{3}(e^{3x+5})'\\&=&6x^{2}e^{3x+5}+2x^{3}(3e^{3x+5})\\&=&6x^{2}e^{3x+5}+6x^{3}e^{3x+5}.\end{array}}$

Final Answer:
$f'(x)\,=\,6x^{2}e^{3x+5}+6x^{3}e^{3x+5}.$

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