Find Derivative of Integral Calculator

What is the derivative of an integral?

The derivative of an integral is a concept that bridges the two main operations of calculus: differentiation and integration. When you take the derivative of an integral with respect to one of its limits, you're essentially finding how the integral changes as that limit changes.

This operation is particularly useful in physics, engineering, and economics where you often need to find rates of change of quantities that are defined as integrals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration in two parts:

1. The first part states that if a function 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).
2. The second part states that if a function f(x) is continuous on an interval [a, b], then the integral of f(x) from a to b can be evaluated using an antiderivative F(x).

This theorem is the foundation for finding derivatives of integrals and is essential for understanding the relationship between differentiation and integration.

How to find the derivative of an integral

Step-by-Step Process

1. Identify the integral and its limits. The integral should be written in the form \( \int_{a}^{x} f(t) \, dt \), where x is the variable limit.
2. Apply the Fundamental Theorem of Calculus. The derivative of the integral with respect to x is simply the integrand evaluated at x.
3. Simplify the expression if possible.

Example calculation

Let's find the derivative of the integral \( \int_{1}^{x} 3t^2 \, dt \) with respect to x.

1. Identify the integral: \( \int_{1}^{x} 3t^2 \, dt \).
2. Apply the Fundamental Theorem: The derivative is \( 3x^2 \).
3. Simplify: The expression is already simplified.

This shows how the derivative of an integral simplifies to the integrand evaluated at the upper limit.

Common mistakes to avoid

• Assuming the derivative of an integral is always the integrand. This is only true when the upper limit is the variable limit.
• Forgetting that the lower limit must be a constant. If both limits are variable, the derivative becomes more complex.
• Miscounting the limits when applying the Fundamental Theorem.

Applications of finding derivatives of integrals

Finding derivatives of integrals has numerous applications in various fields:

• Physics: Calculating velocity from position functions defined as integrals.
• Engineering: Determining rates of change of quantities defined by integrals.
• Economics: Finding marginal functions from integral-based models.
• Statistics: Calculating probability densities from cumulative distribution functions.