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) $.
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
$\displaystyle \chi$ $\displaystyle \sigma$