Derivative Of Integral With X In Bounds

Leibniz's Rule: Differentiating Integrals with Variable Limits

This article explores Leibniz's rule, a powerful theorem in calculus that allows us to differentiate integrals where the limits of integration are functions of the variable with respect to which we're differentiating. Understanding this rule is crucial for solving various problems in physics, engineering, and other fields involving integration and differentiation. We'll delve into the theorem, provide a detailed proof, and illustrate its application with examples.

What is Leibniz's Rule?

Leibniz's rule, also known as the Leibniz integral rule, provides a way to find the derivative of a definite integral whose limits of integration are functions of the variable of differentiation. The rule states that if we have a function defined as:

F(x) = ∫[g(x), h(x)] f(t, x) dt

where f(t, x) is a continuous function of both t and x, and g(x) and h(x) are differentiable functions of x, then the derivative of F(x) with respect to x is given by:

F'(x) = f(h(x), x) * h'(x) - f(g(x), x) * g'(x) + ∫[g(x), h(x)] ∂f(t, x)/∂x dt

This formula elegantly combines the fundamental theorem of calculus with the chain rule. Let's break down each component:

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

• f(g(x), x) * g'(x): This term accounts for the change in the integral due to the variation in the lower limit of integration, g(x). It's the integrand evaluated at the lower limit multiplied by the derivative of the lower limit (note the negative sign).

• ∫[g(x), h(x)] ∂f(t, x)/∂x dt: This term represents the change in the integral due to the dependence of the integrand f(t, x) on x. It's the integral of the partial derivative of the integrand with respect to x, evaluated over the interval [g(x), h(x)].

Proof of Leibniz's Rule (Simplified):

A rigorous proof involves advanced calculus concepts. However, we can outline a simplified intuition:

Imagine the integral as an area under a curve. As x changes, both the limits of integration and the shape of the curve itself can change, affecting the total area. Leibniz's rule accounts for these three effects: the change in area due to the movement of the upper limit, the change due to the movement of the lower limit, and the change due to the deformation of the curve itself.

Examples:

Let's illustrate Leibniz's rule with a few examples:

Example 1: Simple Case

Find the derivative of F(x) = ∫[0, x] t² dt.

Here, g(x) = 0, h(x) = x, and f(t, x) = t². Applying Leibniz's rule:

F'(x) = x² * 1 - 0² * 0 + ∫[0, x] 0 dt = x²

This matches the result obtained by directly evaluating the integral and then differentiating.

Example 2: More Complex Case

Find the derivative of F(x) = ∫[x, x²] sin(tx) dt.

Here, g(x) = x, h(x) = x², and f(t, x) = sin(tx). The partial derivative of f(t, x) with respect to x is t*cos(tx). Applying Leibniz's rule:

F'(x) = sin(x³)*2x - sin(x²) + ∫[x, x²] t*cos(tx) dt

This demonstrates how Leibniz's rule handles cases with both variable limits and an integrand that depends on x.

Conclusion:

Leibniz's rule is a fundamental tool for differentiating integrals with variable limits. It elegantly combines several calculus concepts, providing a powerful method for solving complex problems in various scientific and engineering disciplines. Mastering this rule enhances your ability to tackle challenging integration and differentiation problems effectively. Remember to always carefully identify the functions g(x), h(x), and f(t,x) before applying the rule.