Product rule

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Product rule in calculus is a fundamental rule that allows for the differentiation of products of functions. It is one of the basic rules of differential calculus and is used extensively in mathematical, physical, and engineering problems. The product rule provides a way to find the derivative of a product of two or more functions that are functions of the same variable.

Statement of the Product Rule

The product rule states that if f(x) and g(x) are both differentiable functions, then the derivative of their product with respect to x is given by:

(fg)'(x) = f'(x)g(x) + f(x)g'(x)

where:

• f'(x) and g'(x) are the derivatives of f(x) and g(x) respectively,
• (fg)'(x) is the derivative of the product f(x)g(x).

Application

The product rule is applied in various fields such as physics, engineering, and economics where the product of two variable quantities needs to be differentiated. For example, in physics, the product rule is used to derive the Leibniz rule for differentiation under the integral sign, which is a crucial technique in calculus of variations.

Proof

The proof of the product rule can be approached from the definition of the derivative and applying limit properties. It involves considering the difference quotient of the product f(x)g(x) and applying the limit as x approaches a point.

Examples

1. Given f(x) = x^2 and g(x) = e^x, find the derivative of f(x)g(x). Using the product rule:

(fg)'(x) = 2xe^x + x^2e^x

2. If f(x) = sin(x) and g(x) = cos(x), then the derivative of f(x)g(x) is:

(fg)'(x) = cos^2(x) - sin^2(x)

See Also

• Chain rule
• Quotient rule
• Derivative

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