does anyone know how to take a derivative of a function with respect to a variable if that function contains an integral over another variable? For example, what would be the derivative of the following function with respect to x?
$$f = \int (2x^2y)dy$$
I think it would be
$$f’(x) =\int (4xy)dy$$
but I haven’t found any rule according which it should be so… just used intuition.
Any formal explanation or link to such would be very appreciated. Thanks!
Best Answer
What you surmised is in fact true, because inside the integral, $x$ is just some constant, so for any $a,b$ $$\int_{y=a}^b (2x^2 y) dy =2x^2 \int_{y=a}^b y\, dy $$ and you merely differentiated that (the constant now is the integral) with respect to $x$.
A closely related but less trivial technique is called "differentiation under the integral sign." This says that $$ \frac{d}{dx} \left[ \int_{y=a}^b f(x,y) dy \right] = \int_{y=a}^b \frac{\partial f(x,y)}{\partial x} dy$$ That holds for all $f(x,y)$ that are sufficiently well behaved (certainly holds it it has a continuous derivative, it may hold even with weaker constraints).