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Math Help - Differentiation Hyperbolic Function

  1. #1
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    Differentiation Hyperbolic Function

    Differentiate y=\mathrm{sinh}^{n-1}x\mathrm{cosh}x with respect to x.

    Thanks in advance.
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Air View Post
    Differentiate y=\mathrm{sinh}^{n-1}x\mathrm{cosh}x with respect to x.

    Thanks in advance.
    y'=\sinh^{n-1}(x)\cdot\left(\cosh(x))'+\cosh(x)\cdot(\sinh^{n-1}(x)\right)'

    \left(\cosh(x)\right)'=\left(\frac{e^x+e^{-x}}{2}\right)'=\frac{e^x-e^{-x}}{2}=\sinh(x)

    and \left(\sinh^{n-1}(x)\right)'=(n-1)\left(\sinh^{n-2}(x)\right)\cdot\left(\frac{e^x-e^{-x}}{2}\right)'= (n-1)\sinh^{n-2}(x)\cdot\frac{e^x+e^{-x}}{2}=(n-1)\sinh^{n-2}(x)\cdot\cosh(x)
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  3. #3
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    Hello, Air!

    Differentiate y\:=\: \sinh^{n-1}\!x\cdot \cosh x

    Product Rule: . y' \;=\;\sinh^{n-1}\!x\cdot\sinh x + (n-1)\sinh^{n-2}\!x\cdot\cosh x

    . . . . . . . . . . y' \;=\;\sinh^n\!x + (n-1)\sinh^{n-2}\!x\cdot\cosh x

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  4. #4
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    Quote Originally Posted by Soroban View Post
    Hello, Air!


    Product Rule: . y' \;=\;\sinh^{n-1}\!x\cdot\sinh x + (n-1)\sinh^{n-2}\!x\cdot\cosh x

    . . . . . . . . . . y' \;=\;\sinh^n\!x + (n-1)\sinh^{n-2}\!x\cdot\cosh x

    Isn't it: y' \;=\;\sinh^n\!x + (n-1)\sinh^{n-2}\!x\cdot\cosh ^2 x ?
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Air View Post
    Isn't it: y' \;=\;\sinh^n\!x + (n-1)\sinh^{n-2}\!x\cdot\cosh ^2 x ?
    Yes
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  6. #6
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    Hello, Air!

    Isn't it: y' \;=\;\sinh^n\!x + (n-1)\sinh^{n-2}\!x\cdot\cosh ^2 x ?

    Yes . . . Sorry!

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