The Product Rule

Business Calculus · Axiom Academy

Differentiating products of functions What if we need to differentiate a product of two functions, like f(x) = x^2 (x^3 + 1) ? The derivative of a product is NOT the product of the derivatives! Let's verify this with a simple example. If f(x) = x x = x^2 : Since 1 2x , we clearly need a different rule! If f and g are differentiable functions, then: The derivative of a product = first × derivative of second + second × derivative of first Using prime notation: (fg)' = f'g + fg' Example: Find the derivative of h(x) = x^2(x^3 + 1) We could also expand first: x^2(x^3+1) = x^5 + x^2 , then differentiate: 5x^4 + 2x . Same answer! Example: Find the derivative of y = (3x + 2)(x^2 - 5x) Click each problem to reveal the solution: Use the product rule when two separate functions are being multiplied . If you can easily expand first (like x^2 x^3 = x^5 ), that's often faster! Find the derivative of f(x) = (2x + 3)(x^2 - 1) You've mastered the Product Rule! "First times derivative of second, plus second times derivative of first" Next up: The Quotient Rule for dividing functions!

This is the written version of the interactive lesson above. See the full Business Calculus course.