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.
Calculator
Adjust values & calculateFunction Values at x = 2
Formula
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.
Last reviewed: December 2025
Worked Examples
Example 1: Polynomial Product Derivative
Example 2: Exponential-Polynomial Product
Background & Theory
The Product Rule Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Product Rule Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
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.
How is the product rule used in integration by parts?
Integration by parts is the integral version of the product rule. Starting from d/dx[f*g] = f'g + fg', integrating both sides gives f*g = integral(f'g dx) + integral(fg' dx). Rearranging: integral(fg' dx) = f*g - integral(f'g dx), or equivalently integral(u dv) = uv - integral(v du). This transforms one integral into another that is hopefully simpler. The strategy is to choose u (to differentiate) and dv (to integrate) such that the resulting integral v du is easier than the original. Common applications include integrating x*e^x, x*sin(x), ln(x), and e^x*sin(x). The LIATE rule (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) suggests which function to choose as u.
What is elasticity in economics and how does it use the product rule?
Elasticity measures the percentage change in one quantity relative to the percentage change in another, computed as E = (x/y)*(dy/dx). When y is a product of functions (common in economic models like revenue = price * quantity), the product rule is essential for finding dy/dx. For instance, total revenue R = P(Q)*Q where P depends on quantity Q. Then dR/dQ = P'(Q)*Q + P(Q)*1 by the product rule, and marginal revenue analysis requires this derivative. The elasticity of the product equals the sum of the elasticities of the factors, which is the economic interpretation of the logarithmic derivative formula. This connection makes the product rule indispensable in microeconomics, demand analysis, and pricing theory.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy