Skip to main content

How to Calculate Chain Rule

What is Chain Rule?

The Chain Rule is a formula for differentiating composite functions — functions of the form h(x) = f(g(x)). It is one of the most important rules in calculus and is essential for differentiating all but the simplest functions. The rule states: h'(x) = f'(g(x)) × g'(x).

Formula

In Leibniz notation: dy/dx = (dy/du) · (du/dx)
In
(dy/du) · (du/dx) — (dy/du) · (du/dx)

Step-by-Step Guide

  1. 1If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x)
  2. 2In Leibniz notation: dy/dx = (dy/du) · (du/dx)
  3. 3Identify the outer function f and inner function g
  4. 4Differentiate the outer function with respect to the inner, then multiply by the derivative of the inner

Worked Examples

Input
h(x) = sin(x²), outer=sin, inner=x²
Result
h'(x) = cos(x²) · 2x
Outer derivative: cos(u); inner derivative: 2x
Input
h(x) = (3x+1)⁵
Result
h'(x) = 5(3x+1)⁴ · 3 = 15(3x+1)⁴
Input
h(x) = e^(x²)
Result
h'(x) = e^(x²) · 2x
Input
h(x) = ln(cos(x))
Result
h'(x) = (1/cos(x)) · (−sin(x)) = −tan(x)

Frequently Asked Questions

What is Chain Rule?

The Chain Rule is a formula for differentiating composite functions — functions of the form h(x) = f(g(x)). It is one of the most important rules in calculus and is essential for differentiating all but the simplest functions

How accurate is the Chain Rule calculator?

The calculator uses the standard published formula for chain rule. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.

What units does the Chain Rule calculator use?

This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.

What formula does the Chain Rule calculator use?

The core formula is: If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). Each step in the calculation is shown so you can verify the result manually.

Ready to calculate? Try the free Chain Rule Calculator

Try it yourself →

Settings

PrivacyTermsAbout© 2026 PrimeCalcPro