Chain Rule Calculator — Step-by-Step Derivative Solutions

Calculate derivatives using the chain rule with detailed step-by-step breakdowns. Supports single-variable composite functions, multivariable dz/dt chain rule, and partial derivative chain rule calculations.

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Chain Rule Derivative Calculator

Select your function types and enter coefficients to compute the derivative using the chain rule with full step-by-step working.

Select functions and enter coefficients, then click Calculate Derivative to see the chain rule result.

Chain Rule Formula Explained

The chain rule is a fundamental differentiation technique for composite functions. It states that the derivative of a composite function equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Basic: d/dx[f(g(x))] = f'(g(x)) · g'(x)
Multivariable: dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

Variable Definitions

  • f(u) — The outer function (e.g., sin, cos, e^u, ln, u^n)
  • g(x) — The inner function (e.g., ax+b, ax²+bx+c)
  • f'(u) — Derivative of the outer function with respect to u
  • g'(x) — Derivative of the inner function with respect to x
  • ∂f/∂x, ∂f/∂y — Partial derivatives for multivariable functions

How to Apply the Chain Rule Step by Step

  1. Identify the composite structure — Recognize the outer function f(u) and inner function g(x) where the original function is f(g(x)).
  2. Differentiate the outer function — Compute f'(u), treating the inner function as a single variable u.
  3. Differentiate the inner function — Compute g'(x), the derivative of the inner expression.
  4. Evaluate f'(g(x)) — Substitute g(x) back into f'(u) to get the outer derivative evaluated at the inner function.
  5. Multiply — The final derivative is f'(g(x)) · g'(x).

Chain Rule Calculator Examples

Example 1: sin(ax + b)

Differentiate sin(3x + 2) using the chain rule.

f(u) = sin(u) → f'(u) = cos(u)
g(x) = 3x + 2 → g'(x) = 3
f'(g(x)) = cos(3x + 2)
Result: d/dx[sin(3x+2)] = 3·cos(3x+2)

Example 2: e^(ax² + bx + c)

Differentiate e^(x² + 4x + 1) using the chain rule.

f(u) = e^u → f'(u) = e^u
g(x) = x² + 4x + 1 → g'(x) = 2x + 4
f'(g(x)) = e^(x² + 4x + 1)
Result: (2x + 4)·e^(x² + 4x + 1)

Example 3: Multivariable dz/dt

Find dz/dt when ∂f/∂x = 2, dx/dt = 3t², ∂f/∂y = 4, dy/dt = cos(t).

dz/dt = (2)(3t²) + (4)(cos(t))
dz/dt = 6t² + 4·cos(t)

Real-World Chain Rule Applications

  • Physics & Engineering: Analyzing rates of change in related-rates problems, such as how fluid flow rate changes with pipe diameter and pressure gradients.
  • Economics: Computing how production output changes when multiple input factors (labor, capital) change simultaneously.
  • Machine Learning: Backpropagation in neural networks relies fundamentally on the multivariable chain rule to compute gradients through layers.
  • Biology: Modeling population growth where birth and death rates depend on environmental factors that themselves change over time.
  • Chemistry: Reaction rate analysis where concentration changes depend on temperature, which depends on time.
  • Finance: Option pricing models where underlying asset volatility changes with market conditions.

People Also Ask

The chain rule formula is d/dx[f(g(x))] = f'(g(x)) · g'(x). It states that the derivative of a composite function equals the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. For multivariable cases, dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt).
Step 1: Identify the outer function f(u) and inner function g(x). Step 2: Differentiate f(u) to get f'(u). Step 3: Differentiate g(x) to get g'(x). Step 4: Substitute g(x) into f'(u) to get f'(g(x)). Step 5: Multiply f'(g(x)) by g'(x) to obtain the final derivative.
The multivariable chain rule handles functions of several variables. If z = f(x,y) and both x and y depend on t, then dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). Each independent path contributes a product term. For three variables, add (∂f/∂z)(dz/dt). This generalizes to any number of variables.
The reverse chain rule is an integration technique (also called u-substitution). It reverses the chain rule for differentiation: ∫f'(g(x))·g'(x)dx = f(g(x)) + C. It's used when an integrand contains a function and its derivative, allowing substitution to simplify the integral.
Use the chain rule when differentiating a function of a function (composite), such as sin(x²) or e^(3x). Use the product rule when differentiating two multiplied functions, such as x·sin(x). Sometimes both rules are needed—for example, differentiating x·sin(x²) requires the product rule first, then the chain rule for sin(x²).

Frequently Asked Questions

The calculator handles six common outer functions—sin(u), cos(u), tan(u), e^u, ln(u), and u^n—combined with linear (ax+b) or quadratic (ax²+bx+c) inner functions. It also supports multivariable dz/dt chain rule calculations for functions of two variables.
Yes. Every calculation includes a detailed step-by-step breakdown showing the outer function and its derivative, the inner function and its derivative, the evaluation f'(g(x)), and the final multiplication to produce the chain rule result.
Yes. The multivariable dz/dt mode is specifically designed for Calculus 3 chain rule problems. Enter your partial derivatives ∂f/∂x and ∂f/∂y along with the ordinary derivatives dx/dt and dy/dt to compute dz/dt using the multivariable chain rule formula.
The chain rule is a differentiation technique for composite functions. Implicit differentiation uses the chain rule as part of its process—when differentiating y terms with respect to x, the chain rule gives dy/dx factors. The chain rule is the underlying tool; implicit differentiation is an application of it.
For partial derivatives, the chain rule sums over all intermediate variables. If z = f(x,y) and x = g(s,t), y = h(s,t), then ∂z/∂s = (∂f/∂x)(∂x/∂s) + (∂f/∂y)(∂y/∂s). Each path from the dependent variable to the independent variable contributes a term.
Yes, the reverse chain rule (u-substitution) is the integration counterpart of the chain rule. If you recognize that an integrand has the form f'(g(x))·g'(x), you can substitute u = g(x) and integrate f'(u)du to get f(u) + C = f(g(x)) + C.

Chain Rule Glossary

Chain Rule

A differentiation rule for composite functions: d/dx[f(g(x))] = f'(g(x)) · g'(x).

Composite Function

A function formed by applying one function to the output of another, written as f(g(x)).

Outer Function

The function f(u) applied last in a composite function; its derivative is computed first in the chain rule.

Inner Function

The function g(x) inside the outer function; its derivative g'(x) multiplies the outer derivative in the chain rule.

Partial Derivative

The derivative of a multivariable function with respect to one variable, treating others as constants.

Multivariable Chain Rule

Extension to functions of several variables: dz/dt = Σ(∂f/∂xᵢ)(dxᵢ/dt) for all intermediate variables.

Reverse Chain Rule

Integration technique (u-substitution) that reverses the chain rule: ∫f'(g(x))g'(x)dx = f(g(x)) + C.

Implicit Differentiation

A technique using the chain rule to differentiate equations where y is not explicitly solved in terms of x.

Editorial Review & Methodology

This chain rule calculator was built and reviewed by the NumbrWiz Editorial Team. The chain rule is a cornerstone of differential calculus, verified against standard calculus textbooks including Stewart, Larson, and Thomas' Calculus, as well as the AP Calculus AB/BC curriculum framework and multivariable calculus standards.

  • Formula verification: Cross-checked against multiple authoritative calculus sources and academic references.
  • Edge case testing: Tested with zero coefficients, negative values, fractional powers, and constant inner functions.
  • UX review: Designed for intuitive function selection with clear coefficient inputs and comprehensive step-by-step breakdown.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational purposes; verify critical calculations independently.

Page last reviewed: May 2026 · NumbrWiz Editorial Team