I'm working on the following problem and got stuck:

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for every fixed $x\in(-1,1)$ the function $g_x(t):= e^{tx}f(t) \in L^1(\mathbb{R})$. Let $\varphi:(-1,1)\rightarrow \mathbb{R}$ be defined as

$$\varphi(x) = \int{e^{tx}f(t)}dt$$

Show that $\varphi$ is differentiable.

My attempt:

Let $h(x,t):(-1,1)\times\mathbb{R} \longrightarrow \mathbb{R}$ $$h(x,t) = e^{tx}f(t)$$

Then, in order to differentiate under the integral sign, we need:

$\bullet\space h(.,t)=e^{tx}f(t)\in L^1(\mathbb{R})$ (which we know by definition)

$\bullet\space h(x,.)\in C(\mathbb{R}) \space\space\space[\frac{d}{dx}\left(e^{xt}f(t)\right)=t\cdot e^{xt}f(t)]$


$\bullet\space \lvert\frac{d}{dx} h(x,t)\rvert=\lvert t\cdot e^{xt}f(t)\rvert < s(t)\space$ for some $s(t)\in L^1(\mathbb{R})$

This last step is where I'm stuck. I can't find such $s(t)$ that dominated my function. I think that if I'm able to do so, then I'd be done. Or is my approach incorrect? Thanks.


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