Derivative Calculator of Integral

The derivative of an integral is a fundamental concept in calculus that connects differentiation and integration. This relationship is formalized by the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse operations.

What is the Derivative of an Integral?

The derivative of an integral is a mathematical operation that finds the rate of change of a function that has been integrated. This concept is crucial in calculus and has important applications in physics, engineering, and economics.

When we take the derivative of an integral, we're essentially asking: "How does the accumulated value of a function change as we vary the upper limit of integration?"

Mathematically: If \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F'(x) = f(x) \).

This relationship shows that the derivative of an integral brings us back to the original integrand function.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration. It consists of two parts:

The first part states that every continuous function has an antiderivative.
The second part, which is directly relevant to our topic, states that the derivative of an integral of a function is the original function itself.

The Fundamental Theorem of Calculus is one of the most important theorems in calculus, connecting two seemingly different operations.

How to Calculate the Derivative of an Integral

Calculating the derivative of an integral involves these steps:

Identify the function to be integrated.
Compute the definite integral from a lower limit to an upper limit x.
Differentiate the resulting function with respect to x.

The result will be the original integrand function, demonstrating the inverse relationship between integration and differentiation.

Step	Operation	Result
1	Integrate \( f(t) \) from \( a \) to \( x \)	\( F(x) = \int_{a}^{x} f(t) \, dt \)
2	Differentiate \( F(x) \) with respect to \( x \)	\( F'(x) = f(x) \)

Examples

Example 1: Simple Polynomial

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

1. Compute the integral: \( \int_{0}^{x} 2t \, dt = t^2 \Big|_{0}^{x} = x^2 - 0 = x^2 \)

2. Differentiate the result: \( \frac{d}{dx}(x^2) = 2x \)

The derivative of the integral brings us back to the original function \( 2x \).

Example 2: Trigonometric Function

Now let's consider \( f(x) = \sin(x) \) integrated from 0 to \( x \).

1. Compute the integral: \( \int_{0}^{x} \sin(t) \, dt = -\cos(t) \Big|_{0}^{x} = -\cos(x) + \cos(0) = -\cos(x) + 1 \)

2. Differentiate the result: \( \frac{d}{dx}(-\cos(x) + 1) = \sin(x) \)

Again, the derivative of the integral returns the original function \( \sin(x) \).

FAQ

Why is the derivative of an integral important?

The derivative of an integral is important because it demonstrates the inverse relationship between differentiation and integration, which is fundamental to calculus. It also has practical applications in physics, engineering, and economics.

Can I use this calculator for any function?

This calculator demonstrates the concept using simple functions. For complex functions, you would need to perform the integration and differentiation manually or use more advanced mathematical software.

What happens if the function is not continuous?

The Fundamental Theorem of Calculus requires the function to be continuous on the interval of integration. If the function has discontinuities, the derivative of the integral may not equal the original function.