According to the fundamental theorem of calculus, the first partial derivative is f(x,y).
I'm wondering why I can't apply L'Hopital's rule in the following reasoning:
$\lim_{h\to0}\frac{\int_a^{x+h}f(t,y)dt - \int_a^x f(t,y)dt}{h}=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{1}=0$
While the correct argument should be:
$\begin{align*}\lim_{h\to0}\frac{\int_a^{x+h}f(t,y)dt - \int_a^x f(t,y)dt}{h}&=\lim_{h\to 0}\frac{\int_a^{x+h}f(t,y)dt+\int_x^a f(t,y)dt}{h}=\lim_{h\to 0}\frac{\int_x^{x+h}f(t,y)dt}{h}\\&=\lim_{h\to 0}\frac{f(c,y)h}{h}=\lim_{h\to 0}f(c,y)=f(x,y)\end{align*}$ where $c\in [x,x+h].$