Derivative of An Integral Calculator

This calculator helps you find the derivative of an integral using the Fundamental Theorem of Calculus. Whether you're studying calculus or need to verify your work, this tool provides step-by-step solutions and explanations.

What is the derivative of an integral?

The derivative of an integral is a fundamental concept in calculus that connects differentiation and integration. It's based on the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations.

When you take the derivative of an integral, you're essentially finding the rate of change of the accumulated quantity represented by the integral. This operation is particularly useful in physics, engineering, and economics where rates of change are frequently analyzed.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus has two parts:

The first part states that if a function f is continuous on the closed interval [a, b], and F is the integral of f from a to x, then the derivative of F with respect to x is f(x).
The second part states that if a function f is continuous on the closed interval [a, b], then the integral from a to b of f(x) dx can be found by evaluating an antiderivative F evaluated at the endpoints.

First Part of Fundamental Theorem

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

This theorem establishes a deep connection between differentiation and integration, showing that they are inverse operations. It's one of the most important results in calculus with wide-ranging applications in mathematics and its applications.

How to calculate the derivative of an integral

Calculating the derivative of an integral involves several steps:

Identify the integral function and its limits of integration.
Verify that the integrand is continuous on the interval.
Apply the Fundamental Theorem of Calculus to find the derivative.
Simplify the result if possible.

Important Note

The integrand must be continuous on the closed interval [a, b] for the Fundamental Theorem to apply. If the integrand has discontinuities, the theorem may not hold.

For definite integrals, the derivative with respect to a variable in the upper limit is simply the integrand evaluated at that point. For indefinite integrals, the derivative is the integrand itself.

Examples with solutions

Let's look at some examples to illustrate how to find the derivative of an integral.

Example 1: Simple Polynomial

Find the derivative of ∫[1 to x] 2t dt.

Solution:

First, find the antiderivative: ∫2t dt = t² + C
Evaluate from 1 to x: [x²] - [1²] = x² - 1
Take the derivative: d/dx (x² - 1) = 2x

The result is 2x, which matches the original integrand.

Example 2: Trigonometric Function

Find the derivative of ∫[0 to x] sin(t) dt.

Solution:

Find the antiderivative: ∫sin(t) dt = -cos(t) + C
Evaluate from 0 to x: [-cos(x)] - [-cos(0)] = -cos(x) + 1
Take the derivative: d/dx (-cos(x) + 1) = sin(x)

The result is sin(x), which matches the original integrand.

Integral	Derivative
∫[a to x] 3t² dt	3x²
∫[0 to x] e^t dt	e^x
∫[1 to x] ln(t) dt	ln(x)

FAQ

What is the derivative of an integral?: The derivative of an integral is the original function inside the integral, according to the Fundamental Theorem of Calculus.
When can I use the Fundamental Theorem of Calculus?: You can use the Fundamental Theorem when the integrand is continuous on the closed interval [a, b].
What happens if the integrand is discontinuous?: If the integrand has discontinuities, the Fundamental Theorem may not apply, and you'll need to use other methods to find the derivative.
Can I find the derivative of an indefinite integral?: Yes, the derivative of an indefinite integral is simply the integrand itself.
How does this relate to physics problems?: In physics, the derivative of an integral often represents a rate of change, such as velocity from position or current from charge.