Can someone give me a concrete example of using Leibniz rule for evaluating the derivative with respect to x of f(x,y) where the limits of integration are functions of x? Thanks.
The Liebniz rule says (for well-behaved functions) that:
Define $\displaystyle Fa,b)\mapsto \mathbb{R}$ as $\displaystyle F(x) = \int_c^d f(x,y) dy$ where $\displaystyle f$ is a given function.
Then, $\displaystyle F \ '(x) = \int_c^d \partial_1 f(x,y) dy$
For example, let $\displaystyle f(x,y) = x+y$. And let $\displaystyle (a,b) = (0,1)$. And $\displaystyle c=1,d=0$.
This means, $\displaystyle F(x) = \int_0^1 x + y dy = x + \frac{1}{2}$ and so $\displaystyle F'(x) = 1$.
Using Leibniz rule we also get $\displaystyle F'(x) = \int_0^1 \partial_1 f(x,y) dy = \int_0^1 1 dy = 1$.
Sorry, i didn't get that. You took the partial derivative with respect to what? Not familiar with that partial derivative with a subscript 1 notation.
The specific example I was looking for was where the limit of the integration is also a function of either x or y. So, make the upper limit of integration x^2.