Prove that if $\int_0^1 f(yx) dx = g(y)$, so $f(x)=g(x)+xg'(x)$ for any $x$. I'm trying to solve this question:

$f: \mathbb{R} \to \mathbb{R}$ is a continuous function and $g: \mathbb{R} \to \mathbb{R}$ is of $C^1$ class, such that $g(y) = \int_0^1 f(yx) dx$. Prove that $f(x)=g(x)+xg'(x)$ for all $x \in \mathbb{R}$.

But my progress until now was really slow. I've tried to use the Leibniz integral rule and the first mean value theorem for definite integrals, but I had no success. Can somebody help me?

welcome toi MSE .  a hint can be $$f(x)=g(x)+xg'(x)\\
f(x)=x'g(x)+xg'(x)\\f(x)=(xg(x))'\\\to \\\int f(x)=xg(x)+const$$
