Find derivative of an integral with limits

[Fire flame]

I'm in analysis and I'm trying to understand the following.

Homework Statement

g(x) = integral from 0 to x+δ of f(x)dx + integral from x-δ to 0 of f(x)dx

g'(x) = f(x+δ) - f(x -δ)


So how do they get g'(x)?

 

[LCKurtz]

I'm in analysis and I'm trying to understand the following.

Homework Statement

g(x) = integral from 0 to x+δ of f(x)dx + integral from x-δ to 0 of f(x)dx

g'(x) = f(x+δ) - f(x -δ)


So how do they get g'(x)?

By Leibnitz rule for differentiation of an integral as a function of the upper limit.
By the way, it's better to use a dummy variable in the integrand:$$
\frac d {dx}\int_a^x f(t)\, dt = f(x)$$

 

[Fire flame]

So I understand why there isn't an f(a) since the derivative of a constant is zero, but like in my problem one of my limits is zero and since the function isn't given it could be anything, even something like f(x) = 1/x which at zero is undefined, but in my problem it just goes away to zero. Why? I hope you understand what I'm trying to say.

 

[LCKurtz]

So I understand why there isn't an f(a) since the derivative of a constant is zero, but like in my problem one of my limits is zero and since the function isn't given it could be anything, even something like f(x) = 1/x which at zero is undefined, but in my problem it just goes away to zero. Why? I hope you understand what I'm trying to say.

There are hypotheses on Leibnitz's rule. You can't even talk about, for example, things like$$
F(x) = \int_0^x \frac 1 t\, dt$$because the integral is divergent.