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Math Help - Problems

  1. #1
    Newbie
    Joined
    Jul 2010
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    4

    Unhappy Problems

    Hi everybody
    I am having problems with the following:

    1. Find dy/dx for y=(arccosh sqrt X)^3x

    2. Find integral of (sinhx.cosh2x)dx

    Any help will be much appreciated.
    Thanks
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  2. #2
    Senior Member
    Joined
    Jul 2010
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    Vancouver
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    16
    Lets start with number 1.

    First you need the derivative of (f(x))^{3x}.

    We start by

    <br />
\displaystyle y = f(x)^{3x}

    \displaystyle \ln(y) = 3x\ln(f(x))

    Next we take the derivative of both sides to get

    \displaystyle \frac{1}{y}\frac{dy}{dx} = 3 \ln(f(x)) + 3x\frac{1}{f(x)}f'(x)

    \displaystyle \frac{dy}{dx} = 3 y \left(\ln(f(x))+3x\frac{1}{f(x)}f'(x)\right) = 3 f(x)^{3x} \left(\ln(f(x))+3x\frac{1}{f(x)}f'(x)\right) (1)

    In your case f(x) = \arccosh(\sqrt{x}) = \ln(\sqrt{x}+\sqrt{x-1})

    The derivative of that is

    \displaystyle  f'(x) = \frac{1}{\sqrt{x}+\sqrt{x-1}}\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{x-1}}\right)

    What you now need to do is plug this in in expression (1)
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