To find the Partial derivatives of $x$ and $y$.

Question

Given $f(x,y) = \int_x^y \! g(t) \, \mathrm{d}t.$

$g$ is continuous for all $t$.

I need to find partial derivatives w.r.t $x$ and $y$.

Since no function $g$ is given, then i won't be able to integrate and compute partials. I feel i am missing some concept here needed to solve the question. Please Help me to fin what i am missing.

Thanks.

Answer 1

By the Fundamental Theorem of Calculus: $$f(x,y)=\int_x^y g(t) dt=G(y)-G(x)$$ $$\frac{\partial f}{\partial x}=\frac{\partial G(y)}{\partial x}-\frac{\partial G(x)}{\partial x}$$ $$\frac{\partial f}{\partial x}=-\frac{\partial G(x)}{\partial x}=-g(x)$$

Similarly for the partial derivative w.r.t. $y$.