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Thread: Integrate AFTER differentiate exponent to power exponent

  1. #1
    Junior Member
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    Integrate AFTER differentiate exponent to power exponent

    Hi all,

    I need to differentiate the following

    \begin{equation}
    f(t) = \frac{\partial }{\partial t}e^{-ne^{-\lambda t}}
    \end{equation}

    $$\int_{0}^{+Inf} t dt = ?$$
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  2. #2
    MHF Contributor
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    Re: Integrate AFTER differentiate exponent to power exponent

    To take the derivative in the first equation, use the chain rule:

    \frac{\partial}{\partial t}e^{-ne^{-\lambda t}}=\frac{\partial}{\partial t}e^{-n(e^{-\lambda t})}=-ne^{-ne^{-\lambda t}}\frac{\partial}{\partial t}e^{-\lambda t}=n\lambda e^{-ne^{-\lambda{t}}}e^{-\lambda t}

    The integral diverges to infinity:

    \int_0^\infty t\,dt=\lim_{s\to\infty}\int_0^s t\,dt=\lim_{s\to\infty}\frac{1}{2}s^2=\infty

    - Hollywood
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