1. ## 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 $f'(x)$.

2. 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

$\chi$ $\sigma$