Derivative of Integral Calculator

The derivative of an integral is a fundamental concept in calculus that explores the relationship between differentiation and integration. This calculator helps you compute the derivative of an integral function, providing both the result and a visual representation of the function and its derivative.

What is the Derivative of an Integral?

The derivative of an integral is a mathematical operation that combines differentiation and integration. When you take the derivative of an integral, you're essentially finding how the integral changes as its upper limit varies. This concept is particularly useful in physics, engineering, and economics where rates of change are important.

In calculus, the derivative of an integral with respect to a variable is known as the Fundamental Theorem of Calculus, Part 2. This theorem states that if a function f(x) is continuous on the interval [a, b], and F(x) is the integral of f(x) from a to x, then the derivative of F(x) with respect to x is f(x).

Formula

Derivative of Integral Formula

If F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x).

This formula shows that the derivative of an integral function F(x) is simply the original integrand f(x). This is a powerful result that connects differentiation and integration.

How to Calculate the Derivative of an Integral

Identify the integrand function f(x) that you want to integrate.
Compute the integral F(x) = ∫[a to x] f(t) dt.
Take the derivative of F(x) with respect to x to get F'(x) = f(x).

Important Note

The function f(x) must be continuous on the interval [a, x] for the Fundamental Theorem of Calculus to apply.

Example Calculation

Let's find the derivative of the integral of f(x) = 2x from 0 to x.

First, compute the integral: F(x) = ∫[0 to x] 2t dt = t² evaluated from 0 to x = x² - 0² = x².
Now, take the derivative of F(x): F'(x) = d/dx (x²) = 2x.
According to the Fundamental Theorem of Calculus, F'(x) should equal the original integrand f(x), which is 2x. This confirms our calculation is correct.

FAQ

What is the difference between the derivative of an integral and the integral of a derivative?: The derivative of an integral (FTC Part 2) gives back the original integrand function, while the integral of a derivative (FTC Part 1) gives back the original function plus a constant.
Can I use this calculator for any function?: Yes, you can use this calculator for any continuous function. The calculator will compute the derivative of the integral of the function you provide.
What if my function is not continuous?: The Fundamental Theorem of Calculus requires the function to be continuous on the interval. If your function is not continuous, the derivative of the integral may not be defined.
How is this calculator different from a regular derivative calculator?: This calculator specifically computes the derivative of an integral, which is a special case in calculus. A regular derivative calculator computes the derivative of any function.
Can I use this calculator for definite integrals?: Yes, you can use this calculator for definite integrals. The calculator will compute the derivative of the integral from the lower limit to the upper limit.