Product Rule Differentiation

The Product Rule: A Fundamental Technique for Differentiation

Introduction

Overview of the Product Rule

The product rule in Calculus provides a crucial technique for differentiating functions that are expressed as the product of two other functions. It allows us to find the derivative of a product of two functions by breaking it down into a combination of the derivatives of the individual functions.

Explanation

Product Rule Formula

Let f(x) and g(x) be two functions. The product rule states that the derivative of their product, h(x) = f(x) * g(x), is given by: h'(x) = f'(x) * g(x) + f(x) * g'(x) This formula consists of two terms: * The first term, f'(x) * g(x), represents the derivative of f(x) multiplied by g(x). * The second term, f(x) * g'(x), represents the derivative of g(x) multiplied by f(x).

Justification

The product rule can be justified using the limit definition of the derivative: h'(x) = lim(h(x + h) - h(x)) / h = lim((f(x + h) * g(x + h)) - (f(x) * g(x))) / h = lim(f(x + h) * g(x + h) - f(x) * g(x + h) + f(x) * g(x + h) - f(x) * g(x)) / h = lim(f(x + h) * (g(x + h) - g(x)) + g(x) * (f(x + h) - f(x))) / h = lim(f(x + h) * g'(x) + g(x) * f'(x)) / h = f'(x) * g(x) + f(x) * g'(x) This limit evaluation shows that the derivative of the product is indeed given by the product rule formula.

Applications

Using the Product Rule

The product rule finds applications in various areas of Calculus, including: * Evaluating derivatives of functions expressed as products * Simplifying complex derivatives * Solving optimization problems

Example

Consider the function h(x) = 2x^3 * e^x. Using the product rule: h'(x) = 2x^3 * d/dx(e^x) + e^x * d/dx(2x^3) = 2x^3 * e^x + e^x * 6x^2 = 2x^3 * e^x + 6x^2 * e^x = 2xe^x * (x^2 + 3)

Conclusion

The product rule is a fundamental technique in Calculus that enables us to differentiate functions expressed as products of other functions. Understanding and applying the product rule is crucial for solving various problems in Calculus and related fields.

Pinterest

Pinterest