Derivative Of An Integral With Bounds

Leibniz's Rule: Understanding the Derivative of an Integral with Bounds

This article explores Leibniz's Rule, a powerful theorem in calculus that allows us to find the derivative of an integral whose limits of integration are functions of the variable with respect to which we are differentiating. Understanding this rule is crucial for solving many problems in physics, engineering, and other fields that involve dynamic systems. We'll break down the rule, provide examples, and explore some important considerations.

What is Leibniz's Rule?

Leibniz's Rule, also known as the Leibniz integral rule, states that the derivative of a definite integral with respect to its upper and lower bounds can be calculated using a specific formula. It's a fundamental tool for differentiating integrals where the limits themselves are functions of a variable, a scenario frequently encountered when dealing with integrals representing areas or volumes that change over time.

The general form of Leibniz's Rule is:

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

Where:

• ∫(a(x), b(x)) f(t, x) dt: Represents the definite integral of the function f(t, x) with respect to t, from lower bound a(x) to upper bound b(x). Note that both bounds are functions of x.
• f(t, x): Is the integrand, a function of both the integration variable t and the variable with respect to which we're differentiating, x.
• a(x) and b(x): Are the lower and upper limits of integration, respectively, both being functions of x.
• a'(x) and b'(x): Are the derivatives of the lower and upper limits with respect to x.
• ∂f(t, x)/∂x: Represents the partial derivative of the integrand f(t, x) with respect to x, treating t as a constant.

Understanding the Components:

The formula consists of three parts:

1. f(b(x), x) * b'(x): This term accounts for the change in the integral due to the movement of the upper limit of integration. It's the integrand evaluated at the upper limit multiplied by the derivative of the upper limit.

2. -f(a(x), x) * a'(x): This term accounts for the change in the integral due to the movement of the lower limit of integration. It's the negative of the integrand evaluated at the lower limit multiplied by the derivative of the lower limit.

3. ∫(a(x), b(x)) ∂f(t, x)/∂x dt: This term accounts for the change in the integral due to the change in the integrand itself as x changes. It involves taking the partial derivative of the integrand with respect to x and then integrating the result over the original limits.

Example:

Let's consider the integral:

F(x) = ∫(0, x²) sin(t) dt

To find dF(x)/dx using Leibniz's rule:

• a(x) = 0, a'(x) = 0
• b(x) = x², b'(x) = 2x
• f(t, x) = sin(t), ∂f(t, x)/∂x = 0

Applying Leibniz's rule:

dF(x)/dx = sin(x²) * 2x - sin(0) * 0 + ∫(0, x²) 0 dt = 2x * sin(x²)

Special Cases and Considerations:

• Constant Limits: If both a(x) and b(x) are constants, the first two terms vanish, and the derivative simplifies to the integral of the partial derivative of the integrand with respect to x.

• Integrand Independent of x: If the integrand f(t, x) does not depend on x, then the third term vanishes, and the derivative depends solely on the change in the limits of integration.

Leibniz's Rule is a valuable tool for advanced calculus and its applications. Mastering its use expands your ability to tackle complex problems involving integrals with variable limits. Remember to carefully identify the components of the formula and apply them systematically to achieve accurate results.