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Understanding the Chain Rule in Calculus (With Step-by-Step Examples)

  • Writer: Tyler Buffone
    Tyler Buffone
  • Oct 30
  • 2 min read

If you can recognize when one function is “inside” another, the chain rule becomes straightforward. This post explains the idea in plain language, then walks through carefully chosen examples with complete solutions you can follow line by line.


Abstract waves in teal and orange hues form a gradient pattern, with smooth curves creating a calming, fluid motion across the image. No text.

What the chain rule says


When a function is built by plugging one function f(x) into another function g(x), we have a composite function f(g(x)).


If y = f(g(x)), then


dy/dx = f'(g(x)) ⋅ g'(x)


Think of it as two moves:

  1. Differentiate the outside function while keeping the inside untouched.

  2. Multiply by the derivative of the inside function.


How to Spot a Composite Function Quickly


Look for an operation wrapped around an expression:


  • A power applied to an expression (Example A in the image below).

  • A trig, exponential, or log function applied to an expression (Example B in the image below).

  • A root over an expression (Example C in the image below).


Math examples on dark background: A: (3x² + 5)⁴, B: Sin(3x), e^(5x³ - 2x² + 4), ln(4x² + 1), C: √(7x + 1).

The 4-Step Recipe


  1. Name the inside and differentiate it: set u = inside(x) and determine du/dx

  2. Differentiate the outside with respect to u: determine dy/du.

  3. Multiply the outside derivative by the inside derivative: dy/dx = (dy/du) ⋅ (du/dx).

  4. Substitute u back as the original inside.


Example 1: Power of a Polynomial


Differentiate y = (3x² + 5)⁴.


Solution


  1. Inside: Set u = 3x² + 5 so du/dx = 6x.

  2. Outside: y = u⁴ so dy/du = 4u³.

  3. Chain rule: dy/dx = 4u³ ⋅ 6x.

  4. Substitution: dy/dx = 4u³ ⋅ 6x = 4(3x² + 5)³ ⋅ 6x = 24x(3x² + 5).


Example 2: Natural Log of a Polynomial


Differentiate y = ln(4x² + 1) where 4x² + 1 > 0 for all x.


Solution


  1. Inside: Set u = 4x² + 1 so du/dx = 8x.

  2. Outside: y = ln(u) so dy/du = (1 / u).

  3. Chain rule: dy/dx = (1 / u) ⋅ 8x = (8x / u).

  4. Substitution: dy/dx = (8x / u) = (8x / (4x² + 1)).


Example 3: Exponential with an Expression Inside


Handwritten math showing differentiation using the chain rule. Includes steps for inside, outside, and substitution. Dark background.

Example 4: A Root Over an Expression


Math notes on a dark background detail differentiating \(y = \sqrt{7x+1}\) using the chain rule, showing steps for inside, outside, and substitution.


Need more help mastering the chain rule? If you’re in Winnipeg and looking for a tutor, Tutor Advance provides expert one-on-one support!



 
 
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