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Thread: Differentiable function

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

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

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

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

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

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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