Part 1: The Derivative of an Integral
Part 1 of the FTC states that the derivative of the integral of a function is the original function itself.
In mathematical terms (
a
is a constant):
\frac{d}{dx} \left[\int_{a}^{x} f(t) \, dt \right] = f(x)
The integral is the accumulated area under
f(t)
from
a
to
x
.
The derivative of this accumulated area with respect to
x
is the rate at which the accumulated area changes at
x
.
f(x)
is also the rate of change of the accumulated area at
x
.
Therefore, the derivative of the integral of
f(x)
is
f(x)
.
Part 2: The Integral of a Derivative
Part 2 of the FTC states that the integral, from
a
to
b
, of the derivative of a function is equal to the change in the original function.
In mathematical terms:
\int_{a}^{b} f'(x) \, dx = f(b) - f(a)
The derivative of
f(x)
is the rate of change of
f(x)
.
The integral of this rate of change from
a
to
b
is the total change in
f(x)
from
a
to
b
.
Therefore, the integral of the derivative of
f(x)
is the change in
f(x)
from
a
to
b
.