How does derivative of definite integral make sense

Question

Derivative is taken at a point and hence is value at a point. But definite integral is the value over a domain. Then how come derivative of definite integral make sense.

Answer 1

Take a function $f$. Fix a point in its domain, call it $a$. We would like to know what the integral of this function is for different values, i.e. we have a function of $x$ defined by:

$$F(x) = \int_a^xf(t)dt$$

Note that this is a function of the upper limit of integration.

The fundamental theorem of calculus states that the derivative of this function at a given point $x$ is the value of $f$ evaluated at $x$, i.e.

$$F'(x) = \frac{d}{dx} \int_a^xf(t)dt = f(x) $$

Answer 2

If one or both limits are variable then the definite integral is a function and in some cases the derivative does exist. If both limits are constant then the definite integral is a constant therefore the derivative is zero.