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Math Help - Differentiating the inverse

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    Exclamation Differentiating the inverse

    I took notes in class, but am totally stuck now. I need to find the inverse function of f(x)=4x+10x^9 at c= -14
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    Quote Originally Posted by bnation View Post
    I took notes in class, but am totally stuck now. I need to find the inverse function of f(x)=4x+10x^9 at c= -14
    no, you don't need the inverse of f(x) = 4x + 10x^9.

    if g(x) is the inverse of f(x), then f[g(x)] = x

    \frac{d}{dx}(f[g(x)] = x)

    f'[g(x)] \cdot g'(x) = 1

    g'(x) = \frac{1}{f'[g(x)]}

    note that f(-1) = -14

    since g(x) is the inverse, g(-14) = -1

    so ...

    g'(-14) = \frac{1}{f'[g(-14)]}

    finish up and find g'(-14).
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    Quote Originally Posted by bnation View Post
    I took notes in class, but am totally stuck now. I need to find the inverse function of f(x)=4x+10x^9 at c= -14
    Alternatively, remember that for an inverse, the domain and ranges swap. I.e, the x and y values swap.

    So if f(x) = y = 4x + 10x^9 then f^{-1}(x) = x = 4y + 10y^9.

    You can differentiate the inverse using implicit differentiation.

    x = 4y + 10y^9

    \frac{d}{dx}(x) = \frac{d}{dx}(4y + 10y^9)

    1 = \frac{dy}{dx}\,\frac{d}{dy}(4y + 10y^9)

    1 = \frac{dy}{dx}(4 + 90y^8)

    \frac{dy}{dx} = \frac{1}{4 + 90y^8}.

    It'd be really tough to write the derivative in terms of x though, so you don't bother.

    If you're trying to evaluate this derivative at x = -14 then find what y equals there, then substitute the y value into the derivative.
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