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Calculate The Derivative with Definite Integral

Reviewed by Calculator Editorial Team

Calculating the derivative of an integral is a fundamental concept in calculus that connects differentiation and integration. This process is governed by the Fundamental Theorem of Calculus, which establishes a relationship between the two operations. Understanding this relationship is crucial for solving problems in physics, engineering, and economics.

What is the Derivative of an Integral?

The derivative of an integral represents the rate at which the integral changes with respect to a variable. This concept is particularly useful when dealing with functions that are defined as integrals, such as cumulative distributions in probability or work done by variable forces in physics.

Mathematically, if we have a function f(x) defined as an integral from a lower limit a to an upper limit x of some function g(t) dt, then the derivative of f(x) with respect to x is simply g(x). This is a direct consequence of the Fundamental Theorem of Calculus.

Mathematical Representation:

f(x) = ∫[a to x] g(t) dt

f'(x) = g(x)

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 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) = ∫[a to x] f(t) dt for a ≤ x ≤ b, then F is differentiable on (a, b) and F'(x) = f(x).

This theorem is foundational in calculus as it shows that differentiation and integration are inverse operations, with the derivative of an integral returning the original integrand.

How to Calculate the Derivative of an Integral

To calculate the derivative of an integral, follow these steps:

  1. Identify the function that is defined as an integral. It should be in the form f(x) = ∫[a to x] g(t) dt.
  2. Apply the Fundamental Theorem of Calculus, which states that the derivative of the integral is the integrand evaluated at the upper limit.
  3. If the integral has a variable upper limit, the derivative will be the integrand evaluated at that variable.
  4. If the integral has a variable lower limit, the derivative will be the negative of the integrand evaluated at the lower limit.

Important Note: The derivative of an integral with respect to its upper limit is the integrand evaluated at that upper limit. The derivative with respect to the lower limit is the negative of the integrand evaluated at the lower limit.

Example Calculation

Let's consider the function f(x) = ∫[1 to x] 2t dt. We want to find f'(x).

First, we recognize that f(x) is an integral from 1 to x of the function 2t. According to the Fundamental Theorem of Calculus, the derivative of f(x) with respect to x is simply the integrand evaluated at x.

Step-by-Step Solution:

1. f(x) = ∫[1 to x] 2t dt

2. f'(x) = d/dx [∫[1 to x] 2t dt] = 2x

Therefore, the derivative of f(x) with respect to x is 2x.

Common Mistakes to Avoid

When calculating the derivative of an integral, it's easy to make the following mistakes:

  • Forgetting to apply the Fundamental Theorem of Calculus and instead trying to differentiate the integral directly.
  • Confusing the derivative with respect to the upper limit and the lower limit. Remember that the derivative with respect to the lower limit is the negative of the integrand evaluated at the lower limit.
  • Miscounting the limits of integration, especially when dealing with multiple variables.

To avoid these mistakes, carefully review the Fundamental Theorem of Calculus and practice applying it to various examples.

FAQ

What is the derivative of an integral?
The derivative of an integral is the integrand evaluated at the upper limit of the integral, according to the Fundamental Theorem of Calculus.
How do you find the derivative of an integral with a variable lower limit?
The derivative of an integral with respect to its lower limit is the negative of the integrand evaluated at the lower limit.
What is the Fundamental Theorem of Calculus?
The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration, showing that differentiation is the inverse operation of integration.
Can the derivative of an integral be negative?
Yes, the derivative of an integral can be negative if the integrand is negative or if the derivative is taken with respect to the lower limit.