Derivative of The Integral Calculator

What is the Derivative of the Integral?

In calculus, the derivative of an integral is a fundamental concept that connects differentiation and integration. When you take the derivative of a function that was obtained by integrating another function, you're essentially reversing the integration process.

This operation is made possible by the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations. Specifically, the derivative of an integral with variable limits is equal to the integrand evaluated at the upper limit.

This means that if you have a function \( F(x) \) defined as the integral of \( f(t) \) from a constant \( a \) to \( x \), then the derivative of \( F(x) \) with respect to \( x \) is simply \( f(x) \).

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is one of the most important theorems in calculus. It establishes a deep connection between differentiation and integration. There are two parts to this theorem:

1. The first part states that if a function \( f \) is continuous on the closed interval \([a, b]\), and \( F \) is defined by \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F \) is continuous on \([a, b]\), differentiable on \((a, b)\), and \( F'(x) = f(x) \).
2. The second part states that if \( f \) is continuous on \([a, b]\) and \( F \) is any antiderivative of \( f \) on \([a, b]\), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

The first part of the theorem is particularly relevant to the derivative of the integral calculator, as it shows that the derivative of an integral with variable limits is equal to the integrand evaluated at the upper limit.

How to Calculate the Derivative of an Integral

Calculating the derivative of an integral involves applying the rules of differentiation to a function that was obtained through integration. Here's a step-by-step guide:

1. Identify the function \( F(x) \) that was obtained by integrating another function \( f(t) \).
2. Apply the differentiation rules to find \( F'(x) \).
3. According to the Fundamental Theorem of Calculus, \( F'(x) = f(x) \).

For example, if \( F(x) = \int_{0}^{x} \cos(t) \, dt \), then \( F'(x) = \cos(x) \).

Examples of Derivative of the Integral

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

Example 1

Find the derivative of \( F(x) = \int_{1}^{x} 2t \, dt \).

Solution:

1. First, recognize that \( F(x) \) is the integral of \( f(t) = 2t \) from 1 to \( x \).
2. According to the Fundamental Theorem of Calculus, \( F'(x) = f(x) \).
3. Therefore, \( F'(x) = 2x \).

Example 2

Find the derivative of \( G(x) = \int_{0}^{x} e^{t} \, dt \).

Solution:

1. Here, \( G(x) \) is the integral of \( g(t) = e^{t} \) from 0 to \( x \).
2. Applying the Fundamental Theorem of Calculus, \( G'(x) = g(x) \).
3. Thus, \( G'(x) = e^{x} \).

Applications of the Derivative of the Integral

The derivative of the integral has several important applications in mathematics and science:

• In physics, it's used to find the velocity of an object when the position is given as an integral of acceleration.
• In economics, it's used to find the marginal cost or marginal revenue when these quantities are defined as integrals.
• In engineering, it's used to find the rate of change of quantities that are defined as integrals of other quantities.

Understanding the derivative of the integral is essential for solving many problems in these fields.