Partial derivatives of definite integral

Question

I'm probably overlooking something very simple, but I don't see what it is.
I want to determine the partial derivatives of the integral below, and I know the answers should be:

$\frac{d}{dx}(\int_0^x f(yt) \,dt)=f(yx)$
$\frac{d}{dy}(\int_0^x f(yt) \,dt)=f(yx)x$

But when I tried to write out why this was the case again, I got a different answer for the partial derivative with respect to x:

$\frac{d}{dx}(\int_0^x f(yt) \,dt)=\frac{d}{dx}(F(yx)-F(0))=f(yx)y$
$\frac{d}{dy}(\int_0^x f(yt) \,dt)=\frac{d}{dy}(F(yx)-F(0))=f(yx)x$
(Here $F$ denotes the antiderivative of $f$).

What am I doing wrong?

Answer 1

It should be: $$ \frac{\mathrm d}{\mathrm dx} \int_0^x f(yt) \, \mathrm dt = \frac{\mathrm d}{\mathrm dx} \frac{1}{y}F(yx) = f(yx) $$