How To Find Derivative Of An Integral

How to Find the Derivative of an Integral: Leibniz's Rule Explained

Finding the derivative of an integral might seem counterintuitive at first. After all, integration and differentiation are inverse operations. However, thanks to Leibniz's Rule, also known as the Leibniz integral rule, we can efficiently navigate this seemingly paradoxical situation. This article will delve into the intricacies of Leibniz's Rule, providing a clear understanding of its application and showcasing various examples. Understanding this concept is crucial for advanced calculus and various applications in physics and engineering.

Understanding the Fundamental Theorem of Calculus

Before diving into Leibniz's Rule, it's essential to grasp the Fundamental Theorem of Calculus. This theorem establishes the relationship between differentiation and integration. In essence, it states that differentiation undoes integration and vice versa. However, this simplification only holds true for specific scenarios. The Fundamental Theorem of Calculus is divided into two parts. The part most relevant to our discussion is the second part, which states that if F(x) is an antiderivative of f(x), then:

∫_a^x f(t) dt = F(x) - F(a)

This shows the direct connection between integration and the resulting function.

Leibniz's Rule: The Key to Differentiating Integrals

Leibniz's Rule provides a more general approach to finding the derivative of an integral when the limits of integration are functions of x, not just constants. The rule states:

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

Let's break this down:

• f(t, x): The integrand is a function of both the integration variable 't' and 'x'. This is crucial; Leibniz's rule addresses scenarios where the integrand itself depends on x.

• a(x) and b(x): The lower and upper limits of integration are functions of x.

• b'(x) and a'(x): These represent the derivatives of the upper and lower limits with respect to x, respectively.

• ∂f(t, x)/∂x: This is the partial derivative of the integrand with respect to x, treating 't' as a constant.

Examples Illustrating Leibniz's Rule

Let's solidify our understanding with some examples:

Example 1: Simple Case

Find the derivative of:

∫₀^x² t² dt

Here, a(x) = 0, b(x) = x², and f(t, x) = t². Applying Leibniz's rule:

d/dx [∫₀^x² t² dt] = (x²)² * 2x - (0)² * 0 + ∫₀^x² 0 dt = 2x⁵

Example 2: Integrand Dependent on x

Find the derivative of:

∫₁^x x cos(t) dt

Here, a(x) = 1, b(x) = x, and f(t, x) = x cos(t). Note that the integrand depends on x. Applying Leibniz's rule:

d/dx [∫₁^x x cos(t) dt] = x cos(x) * 1 - x cos(1) * 0 + ∫₁^x cos(t) dt = x cos(x) + sin(x) - sin(1)

These examples demonstrate the power and versatility of Leibniz's Rule. Remember to carefully identify the components (f(t, x), a(x), b(x)) before applying the formula.

Conclusion

Leibniz's Rule is a powerful tool for differentiating integrals with variable limits. It elegantly handles situations where the integrand itself depends on the variable of differentiation, extending the fundamental theorem of calculus to more complex scenarios. Mastering Leibniz's Rule is a significant step towards a deeper understanding of calculus and its numerous applications. Remember to practice with various examples to strengthen your understanding and proficiency.