Mathematics · Calculus
Chain Rule Calculator
Combine an outer derivative and inner derivative to evaluate a composite function's derivative at a point.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Apply f′(g(x))g′(x).
- Multiply 6×4=24.
Understand Chain rule
One idea, three depths
Choose how deeply to explain Chain rule
Chain rule: Combine an outer derivative and inner derivative to evaluate a composite function's derivative at a point.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Chain rule to answer this question: combine an outer derivative and inner derivative to evaluate a composite function's derivative at a point? Enter Outer derivative f′(g(x)) and Inner derivative g′(x); the calculator shows Composite derivative. For example: If f′(g(x))=6 and g′(x)=4, the composite derivative is 24. The answer tells you Composite derivative.
Age 15Explain it to a 15-year-oldConnect it to the formula
A change in x first changes the inner function, then the outer function; multiplying the two local rates gives the total rate. The rule is d[f(g(x))]/dx=f′(g(x))g′(x). Its input values are Outer derivative f′(g(x)), Inner derivative g′(x), and the main result is Composite derivative. For example: If f′(g(x))=6 and g′(x)=4, the composite derivative is 24.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated chain rule relation over the valid real-number domain stated below. The implemented relation is d[f(g(x))]/dx=f′(g(x))g′(x), evaluated from Outer derivative f′(g(x)), Inner derivative g′(x) to produce Composite derivative. A change in x first changes the inner function, then the outer function; multiplying the two local rates gives the total rate. Evaluate the outer derivative at g(x), not at x, before multiplying by the inner derivative.
Inputs and valid domain
- Outer derivative f′(g(x)) must be a finite real number.
- Inner derivative g′(x) must be a finite real number.
Important boundary: Evaluate the outer derivative at g(x), not at x, before multiplying by the inner derivative.
The formula
d[f(g(x))]/dx=f′(g(x))g′(x)
How the calculator works through it
It substitutes Outer derivative f′(g(x)), Inner derivative g′(x) into the formula and exposes every numerical step above. The main output is Composite derivative, accompanied by Outer derivative factor, Inner derivative factor.
Read the result correctly
The Composite derivative is the direct answer to “combine an outer derivative and inner derivative to evaluate a composite function's derivative at a point.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
If f′(g(x))=6 and g′(x)=4, the composite derivative is 24.
Where this model stops being reliable
Evaluate the outer derivative at g(x), not at x, before multiplying by the inner derivative.
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 Chain rule works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Chain rule uses d[f(g(x))]/dx=f′(g(x))g′(x). 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 Chain rule.
Review this foundation about 7 min
Optional enrichment
- Accumulation and integral notation
Integral notation connects Chain 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 Outer derivative f′(g(x)), Inner derivative g′(x).
- Evaluate the principal relationship: d[f(g(x))]/dx=f′(g(x))g′(x).
- Return Composite derivative and check the domain conditions described above.
Python
from math import *
def chain_rule(a, b) -> float:
return (a * b)
assert abs(chain_rule(6, 4) - 24) < 1e-6 * max(1.0, abs(24))
C
#include <assert.h>
#include <math.h>
double chain_rule(double a, double b) {
return (a * b);
}
int main(void) {
const double expected = 24;
const double actual = chain_rule(6, 4);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double chain_rule(double a, double b) {
return (a * b);
}
int main() {
constexpr double expected = 24;
const double actual = chain_rule(6, 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 chain_rule(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global chain_rule
section .text
chain_rule:
push rbp
mov rbp, rsp
sub rsp, 32
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-16]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = chain_rule(a, b)
result = (a * b);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := (a * b);
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). Chain Rule Calculator. MW SysArc Tools. https://math.mwsysarc.com/calculus/chain-rule
MLA 9
MW SysArc. “Chain Rule Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/calculus/chain-rule. Accessed 4 Sept. 2026.
Chicago 17
MW SysArc. “Chain Rule Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed September 4, 2026. https://math.mwsysarc.com/calculus/chain-rule.
Harvard
MW SysArc (2026) ‘Chain Rule Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/calculus/chain-rule (Accessed: 4 September 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_chain_rule_2026,
author = {{MW SysArc}},
title = {Chain Rule Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/calculus/chain-rule},
note = {Published July 21, 2026; accessed September 4, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Chain Rule Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-09-04
UR - https://math.mwsysarc.com/calculus/chain-rule
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Chain rule do?
Combine an outer derivative and inner derivative to evaluate a composite function's derivative at a point.
How does the Chain rule work?
The calculator applies d[f(g(x))]/dx=f′(g(x))g′(x). A change in x first changes the inner function, then the outer function; multiplying the two local rates gives the total rate.
What can I learn from the Chain 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 .