Product Rule Calculator
Free Product rule Calculator for calculus. Enter values to get step-by-step solutions with formulas and graphs. Enter your values for instant results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x)
The product rule states that the derivative of a product of two functions equals the derivative of the first times the second, plus the first times the derivative of the second. This extends to n functions, where you differentiate one factor at a time while keeping others unchanged, summing all terms.
Worked Examples
Example 1: Polynomial Product Derivative
Problem:Find the derivative of h(x) = 3x^2 * 2x^3 at x = 2.
Solution:f(x) = 3x^2, f'(x) = 6x\ng(x) = 2x^3, g'(x) = 6x^2\nProduct rule: h'(x) = f'(x)*g(x) + f(x)*g'(x)\nh'(x) = 6x * 2x^3 + 3x^2 * 6x^2 = 12x^4 + 18x^4 = 30x^4\nAt x = 2: h'(2) = 30 * 16 = 480\nVerification: h(x) = 6x^5, h'(x) = 30x^4, h'(2) = 30*16 = 480 (confirmed!)\nProduct value: h(2) = 3(4) * 2(8) = 12 * 16 = 192
Result:h'(2) = 480 | Term 1 (f'g): 192 | Term 2 (fg'): 288 | Second derivative h''(2) = 960
Example 2: Exponential-Polynomial Product
Problem:Differentiate h(x) = 2e^(3x) * 5x^2 at x = 1.
Solution:f(x) = 2e^(3x), f'(x) = 6e^(3x)\ng(x) = 5x^2, g'(x) = 10x\nProduct rule: h'(x) = 6e^(3x)*5x^2 + 2e^(3x)*10x\nh'(x) = 30x^2*e^(3x) + 20x*e^(3x) = e^(3x)(30x^2 + 20x)\nAt x = 1: h'(1) = e^3 * (30 + 20) = 20.086 * 50 = 1004.28\nProduct value: h(1) = 2e^3 * 5 = 10e^3 = 200.855\nLogarithmic derivative: h'/h = 3 + 2/1 = 5
Result:h'(1) = 1004.28 | f'g term: 602.57 | fg' term: 401.71 | Log derivative: 5
Frequently Asked Questions
What is the product rule for derivatives and when do you use it?
The product rule states that the derivative of a product of two functions equals the first function times the derivative of the second plus the second function times the derivative of the first: d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x). You use the product rule whenever you need to differentiate the product of two or more functions that both depend on the variable. Common examples include x^2*sin(x), e^x*ln(x), and (x+1)*(x^2-3). The product rule is one of the fundamental differentiation rules alongside the power rule, chain rule, and quotient rule. It extends naturally to products of three or more functions and generalizes to the Leibniz rule for higher-order derivatives.
How do you remember the product rule formula?
Several mnemonics help remember the product rule. The most common is: 'the derivative of the first times the second, plus the first times the derivative of the second.' Some students use 'left d-right plus left right-d' where 'd' means 'take the derivative of.' Another approach: if you think of f*g as two friends, you differentiate one while the other stays the same, then swap who gets differentiated, and add the results. In Leibniz notation, d(fg) = (df)g + f(dg), which looks like distributing the d operator. You can also remember it as analogous to the FOIL method for multiplication, but with derivatives: first-outer plus inner-last becomes first-derivative-second plus first-second-derivative.
How does the product rule extend to three or more functions?
For three functions, the product rule becomes: d/dx[f*g*h] = f'*g*h + f*g'*h + f*g*h'. The pattern is clear: differentiate each function one at a time while keeping the others unchanged, then sum all the terms. For n functions, you get n terms. This can be written compactly as d/dx[f1*f2*...*fn] = sum over i from 1 to n of (product of all fj except fi) times fi'. This generalizes to the Leibniz rule for the n-th derivative of a product, which involves binomial coefficients: the n-th derivative of f*g equals the sum from k=0 to n of C(n,k)*f^(k)*g^(n-k), directly analogous to the binomial theorem for expansion of (a+b)^n.
What is the quotient rule and how does it relate to the product rule?
The quotient rule computes the derivative of f(x)/g(x): d/dx[f/g] = [f'*g - f*g'] / g^2. It can be derived from the product rule by writing f/g as f * g^(-1) and applying the product rule with chain rule: d/dx[f*g^(-1)] = f'*g^(-1) + f*(-1)*g^(-2)*g' = f'/g - f*g'/g^2 = (f'g - fg')/g^2. The quotient rule mnemonic is 'low d-high minus high d-low, square the bottom and away we go.' While mathematically equivalent to the product rule plus chain rule, the quotient rule is a convenient shortcut. Some mathematicians prefer to always use the product rule with negative exponents to avoid memorizing a separate formula.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy