Chain Rule Calculator
Verified Calculation Engine
The online Chain Rule Calculator helps you calculate instantly and solve problems related to Differentiation. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.
Applies chain rule for composite functions: d/dx[f(g(x))] = f'(g(x)) · g'(x).
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Chain rule: derivative of outer function (evaluated at inner) times derivative of inner
Chain rule for differentiating composite functions
Inputs
Results
Worked Examples
Example 1: Basic
Find derivative of (x² + 1)³
- outerFunction: u^3
- innerFunction: x^2+1
- derivative: 3(x²+1)² · 2x = 6x(x²+1)²
- steps: f'(u) = 3u², g'(x) = 2x, so f'(g(x))·g'(x) = 3(x²+1)²·2x
Example 2: Board Level
Find derivative of sin(x²)
- outerFunction: sin(u)
- innerFunction: x^2
- derivative: cos(x²) · 2x = 2x cos(x²)
- steps: f'(u) = cos(u), g'(x) = 2x
About this calculator
Overview
Chain Rule Calculator evaluates Chain Rule for the math / differentiation library. Primary input cue: Outer Function f(u) [required]: Outer function. Method anchor: d/dx[f(g(x))] = f'(g(x)) · g'(x) Chain rule: derivative of outer function (evaluated at inner) times derivative of inner. Chain rule for differentiating composi.
When to use
Reach for Chain Rule Calculator to compare scenarios by changing one input at a time for chain rule.
Inputs explained
- Outer Function f(u) [required]: Outer function
- Inner Function g(x) [required]: Inner function
Formula / method
d/dx[f(g(x))] = f'(g(x)) · g'(x) Chain rule: derivative of outer function (evaluated at inner) times derivative of inner. Chain rule for differentiating composite functions (calculator id: chain-rule-calculator).
Worked example
Worked values: outerFunction = u^3, innerFunction = x^2+1 → derivative = 3(x²+1)² · 2x = 6x(x²+1)², steps = f'(u) = 3u², g'(x) = 2x, so f'(g(x))·g'(x) = 3(x²+1)²·2x
Interpreting results
Read the primary output as chain rule under the method on this page.
Assumptions
- Inputs for `chain-rule-calculator` follow the on-page labels; values/shapes must match this operation.
- Units (when shown) must be consistent across inputs for chain rule.
- Real-number arithmetic is used unless chain rule explicitly needs integers.
Limitations
- Chain Rule Calculator cannot invent missing inputs; incomplete forms stop short of a forged chain rule value.
- Display rounding can differ slightly from a CAS/symbolic exact form of chain rule.
- Complex/branch cuts and undefined points are not always interactively annotated for chain rule.
Important note
Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
d/dx[f(g(x))] = f'(g(x)) · g'(x) Chain rule: derivative of outer function (evaluated at inner) times derivative of inner. Chain rule for differentiating composite functions (calculator id: chain-rule-calculator).
Example Calculation
Worked values: outerFunction = u^3, innerFunction = x^2+1 → derivative = 3(x²+1)² · 2x = 6x(x²+1)², steps = f'(u) = 3u², g'(x) = 2x, so f'(g(x))·g'(x) = 3(x²+1)²·2x
Frequently Asked Questions
What is Chain Rule Calculator?
- Chain Rule Calculator evaluates Chain Rule for the math / differentiation library. Primary input cue: Outer Function f(u) [required]: Outer function. Method anchor: d/dx[f(g(x))] = f'(g(x)) · g'(x) Chain rule: derivative of outer function (evaluated at inner) times derivative of inner. Chain rule for differentiating composi.
How does Chain Rule Calculator work?
- d/dx[f(g(x))] = f'(g(x)) · g'(x) Chain rule: derivative of outer function (evaluated at inner) times derivative of inner. Chain rule for differentiating composite functions (calculator id: chain-rule-calculator).
Why use this Math Numbers calculator?
- Reach for Chain Rule Calculator to compare scenarios by changing one input at a time for chain rule.