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Let the function f(x) have the property that f′(x)=(x+1)/(x−3). If g(x)=f(x^2) find g′(x)

Verified Answer :

wizard123

Solve this using chain rule:
Start with g(x) = f(x^2), Differentiate both sides:
\(g'(x) = f'(x^2) * (x^2)' \\ \\ g'(x) = 2x f'(x^2)\)

Now sub in "x^2" into given f'(x) function:
\(g'(x) = 2x*\frac{ x^2 +1}{x^2 -3}\)

yoodyannapolis

\(g'(x)=2x.\frac{(x^2+1)}{(x^2-3)}\)

 

We know chain rule:

\(\frac{d}{dx} f(g(x))=f'(g(x)).g'(x)\)

We have \(f'(x)=\frac{(x+1)}{(x-3)}\)

And \(g(x) = f(x^2)\)

Differentiating both sides, we get

\(g'(x)= f'(x^2).2x\)

Now, replace x by \(x^2\) in f'(x)

\(f'(x)=\frac{(x^2+1)}{(x^2-3)}\)

 So, \(g'(x)=2x.\frac{(x^2+1)}{(x^2-3)}\)

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