Results 1 to 2 of 2

Math Help - inverse hyperbolic derivative

  1. #1
    Member
    Joined
    Dec 2007
    Posts
    137

    Angry inverse hyperbolic derivative

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

    Solving for f'(x), I have:

    f'(x) = h'(g(x))*g'(x)

    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 \frac{1}{\sqrt{x^2-6x}} \geq 1?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by xxlvh View Post
    So in this question I defined g(x) = \frac{1}{\sqrt{x^2-6x}} and h(x) = arccoshx.
    The function I am then evaluating is f(x) = h(g(x)). It is also given that \color{red}\frac{dy}{dx}arccoshx = ln(x+\sqrt{x^2-1}), x \geq 1. The question asks to find the interval on which f(x) is differentiable.

    Solving for f'(x), I have:

    f'(x) = h'(g(x))*g'(x)

    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 \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 \ln(x+\sqrt{x^2-1}). The derivative of arccosh(x) is 1/\sqrt{x^2-1}. For details, see here. Notice that arccosh(x) is only defined for x\geqslant 1. So for x to be in the domain of h(g(x)) it is necessary that g(x))\geqslant1. As you have concluded, this means finding out where 1/\sqrt{x^2-6x} \geqslant 1 .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Derivative of Inverse of Hyperbolic
    Posted in the Calculus Forum
    Replies: 6
    Last Post: June 18th 2011, 06:20 AM
  2. Derivative of Inverse Hyperbolic Tan
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 21st 2011, 08:36 AM
  3. [SOLVED] Hyperbolic Inverse Equation
    Posted in the Calculus Forum
    Replies: 8
    Last Post: January 23rd 2011, 03:23 PM
  4. Inverse hyperbolic question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 14th 2008, 07:42 AM
  5. hyperbolic functions inverse
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 27th 2006, 03:00 PM

Search Tags


/mathhelpforum @mathhelpforum