# How to differentiate a definite integral?

Jul 22, 2017

It depends upon the definite integral in question.

If you were to differentiate an integral with constant bounds of integration, then the derivative would be zero, as the definite integral evaluates to a constant:

Example:

$\frac{d}{\mathrm{dx}} \setminus {\int}_{0}^{1} \setminus x \setminus \mathrm{dx} = 0$ because ${\int}_{0}^{1} \setminus x \setminus \mathrm{dx} = \frac{1}{2}$

However, if we have a variable bound of integration and we differentiate wrt that variable then we get a very different result due to the fundamental theorem of Calculus.

The FTOC tells us that:

$\frac{d}{\mathrm{dx}} \setminus {\int}_{a}^{x} \setminus f \left(t\right) \setminus \mathrm{dt} = f \left(x\right)$ for any constant $a$

(ie the derivative of an integral gives us the original function back), and is the fundamental theorem that relates Differential calculus and integral calculus .

Example:

$\frac{d}{\mathrm{dx}} \setminus {\int}_{0}^{x} \setminus x \setminus \mathrm{dx} = x$ ..... [A]

As the variable of integration is arbitrary we would normally change the integration variable to something different to avoid confusion, i.e:

${\int}_{0}^{1} \setminus x \setminus \mathrm{dx} = {\int}_{0}^{1} \setminus y \setminus \mathrm{dy} = {\int}_{0}^{1} \setminus t \setminus \mathrm{dt} = \frac{1}{2}$

So we would write [A] as:

$\frac{d}{\mathrm{dx}} \setminus {\int}_{0}^{x} \setminus t \setminus \mathrm{dt} = x$