Product rule - Derivatives

The derivative of the product of two terms is equal to the sum of the product of the first term with the derivative of the second term and the product of the second term with the derivative of the first term, or vice verse.

Product Rule of Derivatives

For example,

D ((2x + 1)(x - 1)) = (2x + 1) * D(x - 1) + (x - 1) * D(2x + 1)

• D (x - 1) is equal to D x - D 1 = 1 - 0 = 1
• D (2x + 1) is equal to D 2x + D 1 = 2 + 0 = 2

D ((2x + 1)(x - 1)) = (2x + 1) * 1 + (x - 1) * 2 = 2x + 1 + x - 2 = 3x - 1

So the derivative of (2x + 1)(x - 1) is 3x - 1. Note that the above derivative could also be simply done by first multiplying (2x + 1)(x - 1) by FOIL (or simple, just multiplying the two binomials) to get 2x^2 - x - 1, and then applying sum rule of derivatives to get its derivative. This is just to illustrate the product rule.

The product rule does not apply on

• A product of and variables, for example, 2x, 10y, e x, π x
• A product of constants, such as 10 * 11

The product rule can be applied on

• A product of variables, such as xy, ab, cd
• A product of expressions, such as (x + 1)(x - 1), or (x^2 + 2x + 3)(4x + 1)(1 - x)
• A product of functions, such as f(x) * g(x)
• A product of trigonometric expressions, such as sin(x)cos(x) or (sin x + 1)(cos x - 1)
• A product of logarithmic expressions, such as log(a) * log(b)
• A product of the combination of any of the above mathematical expressions in any order
• A product of more than two terms, in which case you can apply the product rule to two terms each time

The product rule can also be used instead of the quotient rule the quotient of two terms or expressions is considered as a product of the numerator and reciprocal of the denominator, that is, a/b is considered as a times 1/b. In this case if the denominator is not a single variable or function, but an algebraic expression, then you need to apply the chain rule to get the derivative. This does not mean that the product rule is a substitute for the quotient rule. It is better, simpler and easier to work with the correct rule according to the expression you are differentiating.