Determine the derivative of $(f\circ g)(x)$ where $f(x)=-x^2-8\,x-6$ and $g(x)=9\,x^2+4\,x+3$.

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Lesson Parent: 
The Chain Rule
Problem: 

Determine the derivative of $(f\circ g)(x)$ where $f(x)=-x^2-8\,x-6$ and $g(x)=9\,x^2+4\,x+3$.

Answer: 

$(f\circ g)'(x)=f'(g(x))\,g'(x)=-324\,x^3-216\,x^2-284\,x-56$

SAMPLE SOLUTION

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Solution: 

Recall our lessons on function composition and the Chain Rule.

First let's find f'(x) and g'(x). We know what the derivative of a polynomial is. This gives

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and

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Therefore, the composition

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And finally,

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