Determine $(f\circ g)'(x)$ if $f(x)=(x-1)/(x+1)$ and $g(x)=x^2$.

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Lesson Parent: 
The Chain Rule
Problem: 

Determine $(f\circ g)'(x)$ if $f(x)=(x-1)/(x+1)$ and $g(x)=x^2$.

Answer: 

$(f\circ g)'(x)=\frac{4x}{(x^2+1)^2}$

SAMPLE SOLUTION

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Solution: 

We have already determined that <Sign in to see all the formulas>. By the Quotient Rule we see that

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Let's apply the Chain Rule to find <Sign in to see all the formulas>. With the Chain Rule we first find

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and <Sign in to see all the formulas>. Now place $g$ in for every $x$ in $f'(x)$. This gives <Sign in to see all the formulas>. Now place the formula $g(x)$ in for every $g$ in $f'(g)$. This gives <Sign in to see all the formulas>. Finally,

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Note that we found <Sign in to see all the formulas> two different ways. The first by directly determining <Sign in to see all the formulas> and then calculating its derivative. The second method was by determining $f'(x)$ and $g'(x)$ and then using the Chain Rule to determine <Sign in to see all the formulas>. Both methods came up with the same answer as they should.