# Math Help - Leibniz Rule - quick question

1. ## Leibniz Rule - quick question

I'm working through an example following Leibniz rule...but havning trouble following it:

We differentiate:

using Leibniz's Rule:

So just looking at the first integral,

I don't understand the b'(x)f(x,b(x)) part of the rule. Here b(x) = x so b'(x) = 1. But for f(x,b(x)) this would become f(x,x) right? I'm not sure why this is the same as y(x) though??

2. There are 2 Leibniz's you have to apply:
both will generate:
$
\ldots + e^{x-x} y(x) +\ldots - e^{x-x} y(x) + \ldots$

Quickly applied the rule, and found:

$
1+3\int_{0}^{x} e^{3(x-t)} y(t) dt - 3\int_{x}^{1} e^{3(t-x)} y(t) dt
$