Derivative Calculator

The Chain Rule: Master Composite Functions

The chain rule is one of the most important differentiation rules. You need it whenever functions are “nested” or “composed” — that is, when one function serves as the argument of another function.

⛓️ Formula of the Chain Rule

A composite function $f(x)$ consists of an outer function $u(x)$ and an inner function $v(x)$. It is written as $f(x) = u(v(x))$.

Given the function:

$$ f(x) = u(v(x)) $$

Its derivative is the product of the outer and the inner derivative:

$$ f'(x) = u'(v(x)) \cdot v'(x) $$

💡 Tip: “Outer derivative times inner derivative.” Insert the unchanged inner function back into the outer derivative.

🪜 Application in 4 Simple Steps

With this systematic approach you can solve any chain-rule problem.

  1. Identify the functions: Determine the outer function $u(x)$ and the inner function $v(x)$.
  2. Differentiate separately: Compute $u'(x)$ and $v'(x)$ individually.
  3. Substitute the inner into the outer derivative: Take $u'(x)$ and replace every $x$ in it by the full inner function $v(x)$. This gives $u'(v(x))$.
  4. Multiply: Multiply the result from step 3 by the inner derivative $v'(x)$.

✏️ Examples for Illustration

These examples make the application of the rule clear.

Example 1: Power of a Function

Given the function: $f(x) = (3x + 5)^4$

  • 1. Identify: Outer function: $u(x) = x^4$. Inner function: $v(x) = 3x+5$.
  • 2. Differentiate separately: Outer derivative: $u'(x) = 4x^3$. Inner derivative: $v'(x) = 3$.
  • 3. Substitute: $u'(v(x)) = 4(3x+5)^3$.
  • 4. Multiply: $f'(x) = 4(3x+5)^3 \cdot 3$.

The result is: $f'(x) = 12(3x+5)^3$

Example 2: Sine of a Function

Given the function: $f(x) = \sin(x^2)$

  • 1. Identify: Outer function: $u(x) = \sin(x)$. Inner function: $v(x) = x^2$.
  • 2. Differentiate separately: Outer derivative: $u'(x) = \cos(x)$. Inner derivative: $v'(x) = 2x$.
  • 3. Substitute: $u'(v(x)) = \cos(x^2)$.
  • 4. Multiply: $f'(x) = \cos(x^2) \cdot 2x$.

The result is: $f'(x) = 2x \cdot \cos(x^2)$

Example 3: Exponential Function with a Composite Exponent

Given the function: $f(x) = e^{4x^3 - 2x}$

  • 1. Identify: Outer function: $u(x) = e^x$. Inner function: $v(x) = 4x^3 - 2x$.
  • 2. Differentiate separately: Outer derivative: $u'(x) = e^x$. Inner derivative: $v'(x) = 12x^2 - 2$.
  • 3. Substitute: $u'(v(x)) = e^{4x^3 - 2x}$.
  • 4. Multiply: $f'(x) = e^{4x^3 - 2x} \cdot (12x^2 - 2)$.

The result is: $f'(x) = (12x^2 - 2) \cdot e^{4x^3 - 2x}$