Divergence Product Rule

The Product Rule and the Divergence

Introduction

The product rule is a fundamental rule of differentiation in calculus. It states that the derivative of a product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function.

Derivation

To prove the product rule, we use the limit definition of the derivative:

``` f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h ```

Let f(x) = g(x)h(x). Then,

``` f'(x) = lim (h -> 0) [g(x + h)h(x + h) - g(x)h(x)] / h ``` ``` = lim (h -> 0) [g(x + h)h(x + h) - g(x + h)h(x) + g(x + h)h(x) - g(x)h(x)] / h ``` ``` = lim (h -> 0) [g(x + h)h'(x + h) + h(x)g'(x + h)] ``` ``` = g(x)h'(x) + h(x)g'(x) ```

Therefore, the product rule is proven.

Applications

The product rule is used in a variety of applications, including:

• Finding the derivatives of polynomials
• Solving differential equations
• Computing integrals