Derivative of Integral Function Calculator

This calculator helps you find the derivative of an integral function. Learn how integration and differentiation relate mathematically and when this operation is useful in calculus.

What is the derivative of an integral?

The derivative of an integral function is a fundamental operation in calculus that connects integration and differentiation. When you take the derivative of an integral, you're essentially reversing the process of integration and then differentiating the result.

This operation is particularly useful in physics, engineering, and other sciences where you need to analyze rates of change of integrated quantities. The result often simplifies to the original function under the integral, depending on the limits of integration.

How to calculate the derivative of an integral

To find the derivative of an integral function, follow these steps:

Identify the integral function you want to differentiate
Apply the fundamental theorem of calculus which states that the derivative of an integral from a constant to a variable upper limit is the integrand evaluated at that variable
Simplify the result if possible

The process becomes more complex when dealing with definite integrals with variable limits, but the fundamental theorem provides a clear path to the solution.

The formula

The general formula for the derivative of an integral function is:

d/dx ∫[a(x) to b(x)] f(t) dt = f(b(x)) * b'(x) - f(a(x)) * a'(x)

Where:

f(t) is the integrand
a(x) and b(x) are the lower and upper limits of integration
b'(x) and a'(x) are the derivatives of the upper and lower limits

For definite integrals with constant limits, the derivative simplifies to f(b) - f(a).

Worked example

Let's find the derivative of ∫[0 to x] t² dt:

First, recognize that the upper limit is x and its derivative is 1
The lower limit is 0 and its derivative is 0
Apply the formula: d/dx ∫[0 to x] t² dt = x² * 1 - 0² * 0 = x²

The derivative of this integral function is simply x².

FAQ

When is the derivative of an integral equal to the original function?

The derivative of an integral is equal to the original function when the integral has constant limits. For example, d/dx ∫[a to b] f(t) dt = f(x) if a and b are constants.

What is the difference between the derivative of an integral and the integral of a derivative?

The derivative of an integral (with variable limits) gives the difference between the integrand evaluated at the upper and lower limits. The integral of a derivative (with variable limits) gives the original function minus a constant.

Can the derivative of an integral be negative?

Yes, the derivative of an integral can be negative if the integrand evaluated at the upper limit is less than the integrand evaluated at the lower limit.