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Math Help - Differentiable function

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Differentiable function

    Let  f(x) = \int_{1}^{\infty} \frac{e^{-xy}}{y^3}dy . Show that  f(x) is differentiable on  (0,\infty) and find a formula for  f'(x) .
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  2. #2
    MHF Contributor chisigma's Avatar
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    The function is represented as…

    f(x) = \int_{1}^{\infty} \varphi(x,y) dy

    Now \varphi(x,y) is continous and admits partial derivatives for x \in [0,\infty) and y \in [1,\infty), so that the derivative exists and is…

    f^{'}(x) = \int_{1}^{\infty} \varphi_{x} (x,y)\cdot dy = -\int_{1}^{\infty} \frac{e^{-xy}}{y^{2}}\cdot dy

    Kind regards

    \chi \sigma
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