Mathematics · Calculus
Product Rule Calculator
Evaluate the derivative of a product from f, g and their derivatives at the same point.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- First contribution=2×5=10.
- Second contribution=3×4=12.
- Add them: 10+12=22.
Understand Product rule
One idea, three depths
Choose how deeply to explain Product rule
Product rule: Evaluate the derivative of a product from f, g and their derivatives at the same point.
Age 5 Explain it to a 5-year-old Start with a picture
Think of watching something move or grow. Calculus helps measure tiny changes and how they add together. For example: For f=3, f′=2, g=5 and g′=4, the product derivative is 22. The answer tells you Product derivative.
Age 15 Explain it to a 15-year-old Connect it to the formula
Both factors change, so the derivative adds the contribution from changing f and the contribution from changing g. The rule is (fg)′=f′g+fg′. Its input values are Function value f(x), Derivative f′(x), Function value g(x), Derivative g′(x), and the main result is Product derivative. For example: For f=3, f′=2, g=5 and g′=4, the product derivative is 22.
College Explain it at college level State the model precisely
This calculator evaluates a calculus model over the stated real-valued domain. The implemented relation is (fg)′=f′g+fg′, evaluated from Function value f(x), Derivative f′(x), Function value g(x), Derivative g′(x) to produce Product derivative. Both factors change, so the derivative adds the contribution from changing f and the contribution from changing g. This compact symbolic model covers the displayed function family, not every function, discontinuity or domain restriction. The derivative of a product is not f′g′; retain each original factor once.
Inputs and valid domain
- Function value f(x) must be a finite real value.
- Derivative f′(x) must be a finite real value.
- Function value g(x) must be a finite real value.
- Derivative g′(x) must be a finite real value.
Important boundary: The derivative of a product is not f′g′; retain each original factor once.
The formula
(fg)′=f′g+fg′
How the calculator works through it
It substitutes Function value f(x), Derivative f′(x), Function value g(x), Derivative g′(x) into the formula and exposes every numerical step above. The main output is Product derivative, accompanied by f′g contribution, fg′ contribution.
Read the result correctly
The Product derivative is the direct answer to “evaluate the derivative of a product from f, g and their derivatives at the same point.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
For f=3, f′=2, g=5 and g′=4, the product derivative is 22.
Where this model stops being reliable
This compact symbolic model covers the displayed function family, not every function, discontinuity or domain restriction. In particular, the derivative of a product is not f′g′; retain each original factor once.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Continue with a free textbook
OpenStax reading and academic references
Use the calculator as the worked interaction, then continue into the peer-reviewed textbook context. MW SysArc links to OpenStax; the explanation on this page is original and does not reproduce the book.
Calculus Volume 1
Read OpenStax Calculus: Derivatives and integrationCite this book
- APA 7
- Strang, G., & Herman, E. (2016). Calculus volume 1. OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction
- MLA 9
- Strang, Gilbert, and Edwin Herman. Calculus Volume 1. OpenStax, 2016, https://openstax.org/books/calculus-volume-1/pages/1-introduction.
- Chicago author-date
- Strang, Gilbert, and Edwin Herman. 2016. Calculus Volume 1. Houston, TX: OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction.
OpenStax books are free to read online. Their current reuse licence is CC BY-NC-SA; follow the licence shown on the linked book before redistributing or adapting its content.
Clear answers
Frequently asked questions
What does the Product rule do?
Evaluate the derivative of a product from f, g and their derivatives at the same point.
How does the Product rule work?
The calculator applies (fg)′=f′g+fg′. Both factors change, so the derivative adds the contribution from changing f and the contribution from changing g.
What can I learn from the Product rule?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Function value f(x), Derivative f′(x), Function value g(x), Derivative g′(x).
- Evaluate the principal relationship: (fg)′=f′g+fg′.
- Return Product derivative and check the domain conditions described above.
Python
from math import *
def product_rule(a, b, c, x) -> float:
return ((b * c) + (a * x))
assert abs(product_rule(3, 2, 5, 4) - 22) < 1e-6 * max(1.0, abs(22))
C
#include <assert.h>
#include <math.h>
double product_rule(double a, double b, double c, double x) {
return ((b * c) + (a * x));
}
int main(void) {
const double expected = 22;
const double actual = product_rule(3, 2, 5, 4);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double product_rule(double a, double b, double c, double x) {
return ((b * c) + (a * x));
}
int main() {
constexpr double expected = 22;
const double actual = product_rule(3, 2, 5, 4);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double product_rule(double a, double b, double c, double x)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global product_rule
section .text
product_rule:
push rbp
mov rbp, rsp
sub rsp, 64
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd [rbp-32], xmm3
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-24]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-32]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-48]
addsd xmm0, [rbp-56]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
leave
ret
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Last reviewed 2026-07-21. Calculations tested 2026-07-21.