partial derivative of an integral function

Question

Given the function $f(x,y) = \int^x_y g(t) dt $, $g(t)$ continuous for all $t$.

How to evaluate the partial derivatives of $f$ ?

It is correct to do like this (by FTC) ?:

$f_x$ = $g(x)x'$.

$f_y = g(y)y'$.

Thanks!!!

Answer 1

Let $a $ be a real. then

$$f (x,y)=\int_a^x g (t)dt-\int_a^y g (t)dt $$

$$=G (x)-G (y) $$ with $$G(X)=\int_a^X g (t)dt $$ and $$G'(X)=g (X) $$ since $g $ is continuous at $\mathbb R $.

thus

$$f_x (x,y)=G'(x)=g (x) $$ $$f_y (x,y)=-G'(y)=-g (y) $$

Answer 2

"Leibniz's integral rule", which is a generalization of the Fundamental Theorem of Calculus, states that: $\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x, t)dt= f(x, \beta(x))\frac{d}{dx}\beta(x)- f(x, \alpha(x))\frac{d}{dx}\alpha(x)+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f(x, t)}{\partial x} dt$.

Here, $f(x,t)$ is a function of $t$ only, the upper bound on the integral is just $x$ and the lower bound is just $y$. So the derivative of the integral with respect to $x$ is $g(x)$ and the derivative with respect to y is $-g(y)$.