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 5Explain it to a 5-year-oldStart with a picture
Imagine using Product rule to answer this question: evaluate the derivative of a product from f, g and their derivatives at the same point? Enter Function value f(x), Derivative f′(x), Function value g(x), and 1 other input; the calculator shows Product derivative. For example: For f=3, f′=2, g=5 and g′=4, the product derivative is 22. The answer tells you Product derivative.
Age 15Explain it to a 15-year-oldConnect 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.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated product rule relation over the valid real-number domain stated below. 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. 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 number.
- Derivative f′(x) must be a finite real number.
- Function value g(x) must be a finite real number.
- Derivative g′(x) must be a finite real number.
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
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
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Product rule works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Product rule uses (fg)′=f′g+fg′. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Derivatives as rates of change
Rates of change explain the local behaviour captured or approximated by Product rule.
Review this foundation about 7 min
Optional enrichment
- Accumulation and integral notation
Integral notation connects Product rule to accumulated change, area and continuous totals.
Review this foundation about 6 min
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
MATLAB
function result = product_rule(a, b, c, x)
result = ((b * c) + (a * x));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_, x_] := ((b * c) + (a * x));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
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.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
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 entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Product Rule Calculator. MW SysArc Tools. https://math.mwsysarc.com/calculus/product-rule
MLA 9
MW SysArc. “Product Rule Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/calculus/product-rule. Accessed 4 Sept. 2026.
Chicago 17
MW SysArc. “Product Rule Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed September 4, 2026. https://math.mwsysarc.com/calculus/product-rule.
Harvard
MW SysArc (2026) ‘Product Rule Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/calculus/product-rule (Accessed: 4 September 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_product_rule_2026,
author = {{MW SysArc}},
title = {Product Rule Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/calculus/product-rule},
note = {Published July 21, 2026; accessed September 4, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Product Rule Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-09-04
UR - https://math.mwsysarc.com/calculus/product-rule
N1 - Published July 21, 2026
ER -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.
Last reviewed . Calculations tested .