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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)

Respuesta :

wizard123
Solve this using chain rule:
Start with g(x) = f(x^2), Differentiate both sides:
[tex]g'(x) = f'(x^2) * (x^2)' \\ \\ g'(x) = 2x f'(x^2)[/tex]

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

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

 

We know chain rule:

[tex]\frac{d}{dx} f(g(x))=f'(g(x)).g'(x)[/tex]

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

And [tex]g(x) = f(x^2)[/tex]

Differentiating both sides, we get

[tex]g'(x)= f'(x^2).2x[/tex]

Now, replace x by [tex]x^2[/tex] in f'(x)

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

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

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