Calculate Derivative of A Definite Integral

Calculating the derivative of a definite integral involves applying calculus rules to find how the integral's value changes with respect to a variable. This operation is useful in physics, engineering, and optimization problems where you need to understand how quantities evolve over time or space.

What is the Derivative of a Definite Integral?

The derivative of a definite integral with respect to a variable in the limits is a fundamental operation in calculus. When you take the derivative of an integral, you're essentially finding how the integral's value changes as one of its limits varies.

This operation is particularly useful in physics for problems involving moving boundaries, in engineering for optimization problems, and in probability for density functions.

Formula and Calculation

The derivative of a definite integral with respect to one of its limits can be calculated using the Fundamental Theorem of Calculus. The general formula is:

If \( f(x) \) is continuous on \([a, b]\), then:

\[ \frac{d}{db} \left( \int_{a}^{b} f(x) \, dx \right) = f(b) \]

\[ \frac{d}{da} \left( \int_{a}^{b} f(x) \, dx \right) = -f(a) \]

This means that the derivative of the integral with respect to the upper limit is simply the integrand evaluated at the upper limit, and similarly for the lower limit with a negative sign.

Worked Example

Let's calculate the derivative of the integral \( \int_{1}^{x} e^{t} \, dt \) with respect to \( x \).

Using the formula:

\[ \frac{d}{dx} \left( \int_{1}^{x} e^{t} \, dt \right) = e^{x} \]

This shows that the derivative of the integral of \( e^{t} \) from 1 to \( x \) is simply \( e^{x} \).

Applications

The derivative of a definite integral has several practical applications:

Physics: Used in problems involving moving boundaries or time-dependent systems.
Engineering: Applied in optimization problems where the integral represents a quantity that changes over time or space.
Probability: Used in probability density functions to find the rate of change of cumulative distribution functions.

FAQ

What is the derivative of a definite integral?: The derivative of a definite integral with respect to one of its limits is equal to the integrand evaluated at that limit. For the upper limit, it's positive; for the lower limit, it's negative.
When is the derivative of a definite integral useful?: It's useful in physics for moving boundary problems, in engineering for optimization, and in probability for density functions.
What are the assumptions for taking the derivative of a definite integral?: The integrand must be continuous on the interval, and the limits must be differentiable functions.
Can the derivative of a definite integral be zero?: Yes, if the integrand is zero at the point where you're taking the derivative.
How does the derivative of a definite integral relate to the Fundamental Theorem of Calculus?: The Fundamental Theorem of Calculus connects differentiation and integration, and the derivative of a definite integral is a direct application of this theorem.