Find The Derivative of A Definite Integral Calculator

What is the derivative of a definite integral?

The derivative of a definite integral is a powerful result from calculus that connects differentiation and integration. When you take the derivative of a definite integral with respect to one of its limits, you get back the integrand evaluated at that point.

This concept is known as the Fundamental Theorem of Calculus, Part 2. It states that if you have a function defined as the integral of another function, then the derivative of that function is the original integrand.

This theorem is extremely useful in physics, engineering, and other sciences where you need to relate rates of change to accumulated quantities.

How to find the derivative of a definite integral

Finding the derivative of a definite integral involves applying the Fundamental Theorem of Calculus. Here's a step-by-step process:

1. Identify the definite integral in question.
2. Determine which variable you're differentiating with respect to.
3. If the variable is the upper limit, the derivative is the integrand evaluated at that point.
4. If the variable is the lower limit, the derivative is the negative of the integrand evaluated at that point.
5. If the variable is in the integrand, you'll need to use the Leibniz integral rule.

Examples of finding derivatives of definite integrals

Let's look at some concrete examples to illustrate how this works in practice.

Example 1: Simple polynomial

Consider \( F(x) = \int_{1}^{x} 3t^2 \, dt \). What is \( F'(x) \)?

Using the Fundamental Theorem of Calculus:

\( F'(x) = 3x^2 \)

Example 2: Trigonometric function

Now consider \( G(x) = \int_{0}^{\pi} \sin(t) \, dt \). What is \( G'(x) \)?

Notice that the upper limit is π, not x. The derivative with respect to x is zero because the integrand doesn't depend on x.

\( G'(x) = 0 \)

Example 3: Differentiating with respect to lower limit

What if we have \( H(x) = \int_{x}^{2} e^{t} \, dt \)? What is \( H'(x) \)?

When differentiating with respect to the lower limit, we get the negative of the integrand evaluated at the point.

\( H'(x) = -e^{x} \)

Applications of finding derivatives of definite integrals

The ability to find derivatives of definite integrals has many practical applications in various fields:

• Physics: Calculating rates of change of physical quantities like work or energy
• Engineering: Analyzing systems where quantities depend on accumulated values
• Economics: Modeling consumer surplus or producer surplus
• Biology: Studying population growth rates based on accumulated factors

Understanding this concept allows engineers to design more efficient systems, economists to make better policy decisions, and scientists to model complex natural phenomena more accurately.