Integral depending on a parameter

[ehrenfest]

Homework Statement

Let I(y) = $$\int_0^y f(x,y)dx$$.
Find I'(y).

Homework Equations

The Attempt at a Solution

I have the solution which involves making an auxiliary function $$J(t,u) = \int_0^tF(x,u)dx$$ and expressing I'(y) in terms of the partial derivatives of J(t,u). It uses the FTC and the Multivariable Chain Rule. However, it is extremely confusing and if someone could break it down for me that would be great. :)

 

[HallsofIvy]

You could use the chain rule. Or you could use Leibniz's rule which is slightly more general than that:

$$\frac{d}{dy}\int_{\alpha(y)}^{\beta(y)} F(x,y)dx= \int_{\alpha(y)}^{\beta(y)}\frac{\partial F}{\partial y}dx+ \frac{d\alpha}{dy}F(\alpha(y),y)-\frac{d\beta}{dy}F(\beta(y),y)$$