8. Find the derivative of the function $f(x)={\frac {(3x1)^{2}}{x^{3}7}}$.
You do not need to simplify your answer.
Foundations:

This problem involves some more advanced rules of differentiation. In particular, it requires

The Chain Rule: If $f$ and $g$ are differentiable functions, then

$(f\circ g)'(x)=f'(g(x))\cdot g'(x).$

The Quotient Rule: If $f$ and $g$ are differentiable functions and $g(x)\neq 0$ , then

$\left({\frac {f}{g}}\right)'(x)={\frac {f'(x)\cdot g(x)f(x)\cdot g'(x)}{\left(g(x)\right)^{2}}}.$


Solution:

Note that we need to use chain rule to find the derivative of $\left(3x1\right)^{2}$. Then we find

$f'(x)$ 
$=\,\,{\frac {\left[\left(3x1\right)^{2}\right]'\cdot (x^{3}7)\,\,\,\,\left(3x1\right)^{2}\cdot (x^{3}7)'}{(x^{3}7)^{2}}}$ 

$=\,\,{\frac {\left[2\left(3x1\right)\cdot 3\right]\cdot (x^{3}7)\,\,\,\,\left(3x1\right)^{2}\cdot 3x^{2}}{(x^{3}7)^{2}}}.$ 

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