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Thread: inverse hyperbolic derivative

  1. #1
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    Angry inverse hyperbolic derivative

    So in this question I defined $\displaystyle g(x) = \frac{1}{\sqrt{x^2-6x}} $ and $\displaystyle h(x) = arccoshx$.
    The function I am then evaluating is $\displaystyle f(x) = h(g(x))$. It is also given that $\displaystyle \frac{dy}{dx}arccoshx = ln(x+\sqrt{x^2-1}), x \geq 1.$ The question asks to find the interval on which $\displaystyle f(x)$ is differentiable.

    Solving for f'(x), I have:

    $\displaystyle f'(x) = h'(g(x))*g'(x) $

    $\displaystyle f'(x) = ln(\frac{1}{\sqrt{x^2-6x}} + \sqrt{\frac{1}{x^2-6x}-1})(\frac{x-3}{\sqrt{x^2-6x}}) $

    Hopefully my work up to this point is correct, but to find out where the function is differentiable, do I just need to set $\displaystyle \frac{1}{\sqrt{x^2-6x}} \geq 1? $
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  2. #2
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    Quote Originally Posted by xxlvh View Post
    So in this question I defined $\displaystyle g(x) = \frac{1}{\sqrt{x^2-6x}} $ and $\displaystyle h(x) = arccoshx$.
    The function I am then evaluating is $\displaystyle f(x) = h(g(x))$. It is also given that $\displaystyle \color{red}\frac{dy}{dx}arccoshx = ln(x+\sqrt{x^2-1}), x \geq 1.$ The question asks to find the interval on which $\displaystyle f(x)$ is differentiable.

    Solving for f'(x), I have:

    $\displaystyle f'(x) = h'(g(x))*g'(x) $

    $\displaystyle f'(x) = ln(\frac{1}{\sqrt{x^2-6x}} + \sqrt{\frac{1}{x^2-6x}-1})(\frac{x-3}{\sqrt{x^2-6x}}) $

    Hopefully my work up to this point is correct, but to find out where the function is differentiable, do I just need to set $\displaystyle \frac{1}{\sqrt{x^2-6x}} \geq 1? $
    There's a mistake here. It's the function arccosh(x) itself, not its derivative, that is equal to $\displaystyle \ln(x+\sqrt{x^2-1})$. The derivative of arccosh(x) is $\displaystyle 1/\sqrt{x^2-1}$. For details, see here. Notice that arccosh(x) is only defined for $\displaystyle x\geqslant 1$. So for x to be in the domain of h(g(x)) it is necessary that $\displaystyle g(x))\geqslant1$. As you have concluded, this means finding out where $\displaystyle 1/\sqrt{x^2-6x} \geqslant 1 $.
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