f(x) = 4*x^3 + 2* e^2x is invertible :: Find d/dx (f^-1(2))

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We know by defnition of inverse function.

`f^(-1)(f(x))=x`

Let `g=f^(-1)` ,then

`g(f(x))=x` (i)

differentiating (i) with respect to x ,implicitly

`g'(f(x))f'(x)=1`

`g'(f(x))=1/(f'(x)) ``(ii)`

`given `

`f(x)=4x^3+2e^(2x)`

`f'(x)=12x^2+4e^(2x) `(iii)

`f(x)=2=4x^3+2x^(2x)`

`1=2x^3+e^(2x) `(iv)

one of the solution of (iv) is x=0

so substitute x=0 in (iii) ,we have `f'(0)=4`

Thus from (ii)

`g'(f(x))=g'(2)=1/4`

`f^(-1)'(2)=1/4`

Ans.

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