Determine the derivative of $(f\circ g)(x)$ where $f(x)=-x^2-5$ and $g(x)=x+5$.

Primary tabs

Lesson Parent: 
The Chain Rule
Problem: 

Determine the derivative of $(f\circ g)(x)$ where $f(x)=-x^2-5$ and $g(x)=x+5$.

Answer: 

$(f\circ g)'(x)=f'(g(x))\,g'(x)=-2\,x-10$

SAMPLE SOLUTION

All of our problems have fully worked out, logical solutions. We connect every solution to its underlying lesson so you can understand the concepts needed to solve any problem and reinforce your learning by practicing with similar problems. All of the formulas are available to our subscribers, but here is a sample of the problem solution you will see when you log in:



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

<Sign in to see all the formulas>

and

<Sign in to see all the formulas>

Therefore, the composition

<Sign in to see all the formulas>

And finally,

<Sign in to see all the formulas>