Derivative of Integral Calculator with Bounds

This calculator helps you find the derivative of an integral with bounds. Whether you're studying calculus or need to solve a specific problem, this tool provides both the result and a step-by-step explanation.

What is the Derivative of an Integral?

The derivative of an integral is a fundamental concept in calculus that combines differentiation and integration. When you take the derivative of an integral, you're essentially finding how the integral changes as its upper limit varies.

This operation is particularly useful in physics, engineering, and economics where you need to analyze rates of change of accumulated quantities.

How to Calculate the Derivative of an Integral

Calculating the derivative of an integral with bounds involves applying the Fundamental Theorem of Calculus. Here's a step-by-step process:

Identify the integral function and its bounds.
Differentiate the upper bound of the integral.
Multiply the result by the integrand evaluated at the upper bound.
If the integral has a lower bound that depends on the variable, you'll need to subtract the derivative of the lower bound multiplied by the integrand evaluated at the lower bound.

Our calculator automates this process, but understanding these steps helps you verify the results and apply the concept to more complex problems.

The Formula

The general formula for the derivative of an integral with bounds is:

d/dx ∫[a(x), b(x)] f(t) dt = f(b(x)) * b'(x) - f(a(x)) * a'(x)

Where:

f(t) is the integrand
a(x) and b(x) are the lower and upper bounds of integration
b'(x) and a'(x) are the derivatives of the upper and lower bounds

Worked Example

Let's find the derivative of ∫[x, 2x] t² dt with respect to x.

First, evaluate the integral: ∫ t² dt = (1/3)t³ + C
Now, apply the derivative formula:
d/dx [(1/3)(2x)³ - (1/3)x³] = (2x)² * 2 - x² * 1
Simplify: 4x² - x² = 3x²

The derivative is 3x², which our calculator would confirm.

FAQ

What is the difference between the derivative of an integral and the integral of a derivative?

The derivative of an integral (with respect to the upper bound) gives you the integrand evaluated at that bound, while the integral of a derivative (with respect to the same variable) gives you the original function minus a constant. The Fundamental Theorem of Calculus connects these two operations.

When would I need to calculate the derivative of an integral?

You might need this calculation when analyzing rates of change in physical systems, solving differential equations, or working with probability distributions where you need to find the probability density function from a cumulative distribution function.

Can the calculator handle integrals with variable bounds?

Yes, our calculator can handle integrals with variable bounds. Simply enter the functions for both the upper and lower limits, and it will compute the derivative correctly using the formula shown above.