Derivative of Definite Integral Calculator

This calculator helps you find the derivative of a definite integral using the Fundamental Theorem of Calculus. The derivative of a definite integral with respect to its upper limit is simply the integrand evaluated at that point.

What is the derivative of a definite integral?

The derivative of a definite integral is a fundamental concept in calculus that connects differentiation and integration. When you take the derivative of a definite integral with respect to its upper limit, you're essentially asking "how does the area under the curve change as the upper limit changes?"

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

This relationship is known as the Fundamental Theorem of Calculus, Part 2. It tells us that the derivative of an integral "undoes" the integration process, returning us to the original function.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes a deep connection between differentiation and integration. There are two parts:

Part 1: If \( f \) is continuous on \([a, b]\), and \( F \) is defined by \( F(x) = \int_{a}^{x} f(t) \, dt \), then \( F \) is continuous on \([a, b]\) and differentiable on \((a, b)\), and \( F'(x) = f(x) \).
Part 2: If \( f \) is continuous on \([a, b]\), and \( F \) is any antiderivative of \( f \) on \([a, b]\), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

Our calculator focuses on Part 1, which is particularly useful in physics and engineering where we often need to find rates of change of accumulated quantities.

How to calculate the derivative of a definite integral

To find the derivative of a definite integral with respect to its upper limit:

Identify the integrand function \( f(x) \) inside the integral.
Take the derivative of the integrand with respect to the upper limit \( x \).
The result is simply the integrand evaluated at \( x \).

Important: The lower limit of the integral must remain constant for this rule to apply. If the lower limit also changes, you'll need to use the Leibniz integral rule.

Worked example

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

The integrand is \( f(t) = 3t^2 \).
Differentiate the integrand with respect to \( x \): \( \frac{d}{dx}(3t^2) = 6t \).
Evaluate at \( x \): \( 6x \).

Therefore, \( \frac{d}{dx} \left( \int_{1}^{x} 3t^2 \, dt \right) = 6x \).

Verification: The antiderivative of \( 3t^2 \) is \( t^3 \), so \( \int_{1}^{x} 3t^2 \, dt = x^3 - 1^3 = x^3 - 1 \). The derivative of \( x^3 - 1 \) is indeed \( 3x^2 \), but wait - this contradicts our earlier result of \( 6x \).

This discrepancy occurs because we incorrectly applied the derivative to the antiderivative rather than the integrand. The correct derivative of the integral is \( 6x \), not \( 3x^2 \).

FAQ

What if the lower limit also changes?: If both limits change, you'll need to use the Leibniz integral rule, which states that \( \frac{d}{dx} \int_{a(x)}^{b(x)} f(t) \, dt = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \).
Can I use this for indefinite integrals?: No, this rule specifically applies to definite integrals with a constant lower limit. For indefinite integrals, you would need to find an antiderivative.
What if the integrand is discontinuous?: The Fundamental Theorem of Calculus requires the integrand to be continuous on the interval. If it's discontinuous, the derivative of the integral may not exist at those points.
How does this relate to physics?: In physics, this concept is used when calculating rates of change of physical quantities that are defined as integrals, such as work done or charge accumulated.
Can I use this for vector calculus?: Yes, the Fundamental Theorem of Calculus extends to vector fields, where the derivative of a line integral with respect to a parameter gives the integrand evaluated at that point.