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Math Help - inverse function derivative

  1. #1
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    inverse function derivative

    what is the value of (d/dx)[f^(-1) (x) ] when x=-2 given that f(x)=x^3+x.
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  2. #2
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    Quote Originally Posted by gracy View Post
    what is the value of (d/dx)[f^(-1) (x) ] when x=-2 given that f(x)=x^3+x.
    First of all f(x) is an increasing function of x so the inverse function exists on \mathbb{R}

    Let f(y)=x, then

    \frac{d}{dx}f(y)=f'(y)\frac{dy}{dx}=(3y^2+1)\frac{  dy}{dx}=1.

    Hence:

    \frac{dy}{dx}=\frac{1}{3y^2+1}=\frac{1}{3\,f(x)+1}  =\frac{1}{3x^3+3x+1}

    but y=f^{-1}(x), so:

    \frac{d}{dx}f^{-1}(x)=\frac{1}{3x^3+3x+1}.

    So finally:

    \left. \frac{d}{dx}f^{-1}(x)\right|_{x=-2}=\frac{1}{-3\times 8-3 \times 2+1}\approx -0.0345.

    RonL
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