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Math Help - Derivative of an exponential when the exponent is a function

  1. #1
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    Derivative of an exponential when the exponent is a function

    what is the value of
    <br />
\frac{dy}{dx}<br />
    if
    <br />
y=2^{\sin (2^{\sin(2^x)})}<br />
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  2. #2
    Senior Member Twig's Avatar
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    Use the fact that a^{f(x)}=e^{ln(a)^{f(x)}}=e^{f(x)\cdot ln(a)}
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  3. #3
    Super Member Random Variable's Avatar
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     y = 2^{\sin 2^{sin2^{x}}}

    let  u= 2^{x}

    then  y = 2^{\sin 2^{\sin u}}

    let  v = \sin u

    then  y = 2^{\sin 2^{v}}

    let  w = 2^v

    then  y = 2^{\sin w}

    finally let  z = \sin w

    then  y = 2^{z}

     \frac {dy}{dx} = \frac {dy}{dz} \frac {dz}{dw} \frac {dw}{dv} \frac {dv}{du} \frac {du}{dx}

     \frac {dy}{dx} = \ln(2) 2^{z} \cos(w) ln(2)2^{v} \cos(u) ln(2) 2^x

     = \ln(2) 2^{\sin w} \cos(2^{v}) ln(2)2^{\sin u} \cos(2^{x}) \ln(2) 2^{x}

     = \ln(2) 2^{\sin 2^{v}} \cos(2^{\sin u}) \ln(2)2^{\sin 2^{x}} \cos(2^{x}) \ln(2) 2^{x}

     = \ln(2) 2^{\sin 2^{\sin 2^x}} \cos(2^{\sin 2^{x}}) \ln(2)2^{\sin 2^{x}} \cos(2^{x}) \ln(2) 2^{x}
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