Calculating The Derivative of A Definite Integral

Introduction

In calculus, we often need to find the rate of change of a function that is defined as an integral. For example, if we have a function F(x) defined as the integral from a to x of f(t) dt, we might want to find F'(x), the derivative of F with respect to x. This operation is known as differentiating under the integral sign.

The key insight comes from the Fundamental Theorem of Calculus, which tells us that if F(x) is the antiderivative of f(x), then the derivative of F(x) is f(x). When dealing with definite integrals, this principle extends to the upper limit of integration.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus consists of two parts:

1. The first part states that if a function f is continuous on the closed interval [a, b], and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a).
2. The second part, which is relevant to our topic, states that if f is continuous on [a, b], and F is defined by F(x) = ∫ from a to x of f(t) dt, then F is differentiable on (a, b) and F'(x) = f(x).

This second part is what allows us to calculate the derivative of a definite integral.

Derivative Formula

The formula for calculating the derivative of a definite integral is:

This formula works when:

• The integrand f(t) is continuous on the interval [a, x]
• The upper limit of integration is x (the variable with respect to which we're differentiating)
• The lower limit of integration a is a constant

If the lower limit is also a variable, the formula becomes more complex and involves the derivative of the lower limit as well.

Worked Example

Let's work through an example to see how this works in practice.

Consider the function F(x) defined as:

We want to find F'(x), the derivative of F with respect to x.

Using our formula:

This makes sense because the antiderivative of t² is (t³)/3, and evaluating this from 0 to x gives (x³)/3 - (0³)/3 = x³/3. The derivative of x³/3 is x², which matches our result.

Common Mistakes

When calculating the derivative of a definite integral, there are several common mistakes to avoid:

1. Assuming the derivative of the integral is always the integrand: This only works when the upper limit is the variable of differentiation and the lower limit is constant.
2. Forgetting to check the continuity of the integrand: The integrand must be continuous on the interval of integration.
3. Incorrectly differentiating the limits of integration: If the lower limit is also a variable, you need to use the Leibniz rule for differentiation under the integral sign.
4. Ignoring the Fundamental Theorem of Calculus: Not understanding the relationship between differentiation and integration can lead to incorrect results.

Applications

Calculating the derivative of a definite integral has many practical applications:

• In physics, it helps find the velocity of a particle given its position as an integral of acceleration.
• In engineering, it's used to find the rate of change of quantities that are defined as integrals, such as work or energy.
• In economics, it helps analyze the marginal value of additional units of a resource.
• In probability, it's used in the calculation of probability density functions.