The Chain Rule: Complete Mastery Guide
The Chain Rule is arguably the most important derivative rule in calculus. Master it with our comprehensive guide featuring intuitive explanations, 20+ worked examples ranging from basic to advanced, proven shortcuts, and practice problems with detailed solutions.
The Chain Rule appears in approximately 70% of all calculus problems. If you can't recognize and apply it fluently, you'll struggle with derivatives, integrals, optimization, and beyond. This is THE rule to master completely!
📑 Table of Contents
🔗 What is the Chain Rule?
The Chain Rule allows us to differentiate composite functions - functions that are "nested" inside other functions.
Recognizing Composite Functions
A composite function has the form f(g(x)), where one function is inside another function.
- (x² + 1)⁵ → The function x² + 1 is raised to the 5th power
- sin(3x) → The function 3x is inside the sine function
- e^(x²) → The function x² is in the exponent
- √(x³ - 2x) → The function x³ - 2x is inside the square root
d/dx[sin(3x)] = cos(3x) ← WRONG!
The correct answer is: 3cos(3x) (using Chain Rule)
Ask yourself: "Is there a function INSIDE another function?" If yes, you need the Chain Rule!
📐 The Formula & Intuition
The Chain Rule Formula
In Words: "Derivative of the outer function (keeping the inside unchanged) TIMES the derivative of the inner function"
The Step-by-Step Process
Label them clearly: outer = f, inner = g
Find f'(g(x)) - pretend the inside is just a variable
Find g'(x) and multiply it by your result from Step 2
Think of it like unwrapping a gift: First unwrap the outside layer (differentiate outer function), then unwrap the inside layer (differentiate inner function).
🌱 Basic Chain Rule Examples
Let's start with simple examples to build confidence and muscle memory.
Find: d/dx[(x² + 1)⁵]
- Outer function: ( )⁵ (something to the 5th power)
- Inner function: x² + 1
- f(u) = u⁵ → f'(u) = 5u⁴
- f'(g(x)) = 5(x² + 1)⁴
- g(x) = x² + 1 → g'(x) = 2x
- Final answer: 5(x² + 1)⁴ · 2x = 10x(x² + 1)⁴
Find: d/dx[sin(3x)]
- Outer: sin( )
- Inner: 3x
- d/dx[sin(u)] = cos(u)
- cos(3x)
- d/dx[3x] = 3
- Final answer: 3cos(3x)
Find: d/dx[e^(5x)]
Solution: e^(5x) · 5 = 5e^(5x)
Find: d/dx[√(x² + 4)]
Rewrite as (x² + 4)^(1/2)
Solution: (1/2)(x² + 4)^(-1/2) · 2x = x/√(x² + 4)
⚡ Intermediate Chain Rule Examples
Find: d/dx[(2x³ - 5x + 1)⁷]
Solution: 7(2x³ - 5x + 1)⁶ · (6x² - 5) = 7(6x² - 5)(2x³ - 5x + 1)⁶
Find: d/dx[x² · sin(x³)]
Use Product Rule: (first · d/dx[second]) + (second · d/dx[first])
Solution: x² · 3x²cos(x³) + sin(x³) · 2x = 3x⁴cos(x³) + 2xsin(x³)
Find: d/dx[sin²(3x)]
Three layers: ( )², sin( ), 3x
Solution: 2sin(3x) · cos(3x) · 3 = 6sin(3x)cos(3x)
Find: d/dx[cos(2x)/(x² + 1)]
Solution: [−2sin(2x)(x² + 1) − cos(2x)(2x)] / (x² + 1)²
🚀 Advanced Chain Rule Applications
Find: d/dx[e^(sin(x²))]
Three layers: e^( ), sin( ), x²
Solution: e^(sin(x²)) · cos(x²) · 2x = 2xe^(sin(x²))cos(x²)
Find: d/dx[ln(√(x² + 1))]
Simplify: ln(√(x² + 1)) = (1/2)ln(x² + 1)
Solution: (1/2) · [1/(x² + 1)] · 2x = x/(x² + 1)
Find dy/dx: x² + y² = 25
Differentiate both sides (y is function of x, use Chain Rule on y²)
2x + 2y(dy/dx) = 0
Answer: dy/dx = -x/y
⚠️ Common Mistakes to Avoid
Wrong: d/dx[sin(2x)] = cos(2x)
Right: d/dx[sin(2x)] = cos(2x) · 2 = 2cos(2x)
Why: You MUST multiply by the derivative of the inside (2x)!
Wrong: d/dx[(x² + 1)⁵] = 5(2x)⁴ · 2x
Right: d/dx[(x² + 1)⁵] = 5(x² + 1)⁴ · 2x
Why: When differentiating the outer, keep the inside EXACTLY as it was!
Wrong: d/dx[e^(x²)] = x² · e^(x²−1)
Right: d/dx[e^(x²)] = e^(x²) · 2x
Why: For e^(something), derivative is e^(something) times derivative of something!
Problem: d/dx[cos²(3x)]
Wrong: -2sin(3x) (missing one layer)
Right: 2cos(3x) · [-sin(3x)] · 3 = -6cos(3x)sin(3x)
Why: There are THREE layers: ( )², cos( ), and 3x
🎯 Practice Problems
- d/dx[(5x − 2)⁴]
- d/dx[cos(4x)]
- d/dx[e^(−3x)]
- d/dx[√(2x + 5)]
- 20(5x − 2)³
- −4sin(4x)
- −3e^(−3x)
- 1/√(2x + 5)
- d/dx[(x³ + 2x)⁶]
- d/dx[x · e^(x²)]
- d/dx[sin(2x)/x]
- d/dx[ln(x² − 1)]
- 6(3x² + 2)(x³ + 2x)⁵
- e^(x²)(1 + 2x²)
- [2xcos(2x) − sin(2x)]/x²
- 2x/(x² − 1)
- d/dx[e^(2x)·sin(3x)]
- d/dx[√(sin(x²))]
- d/dx[ln(e^x + e^(−x))]
- d/dx[tan³(2x)]
- e^(2x)[2sin(3x) + 3cos(3x)]
- [xcos(x²)]/√(sin(x²))
- (e^x − e^(−x))/(e^x + e^(−x))
- 6tan²(2x)sec²(2x)
🚀 Practice with Our Calculator
Verify your chain rule solutions instantly with step-by-step explanations!
Try Chain Rule Calculator →📚 Key Takeaways
- Recognition: Look for functions inside functions
- Process: Outer derivative × inner derivative
- Practice: Do 20-30 problems to build muscle memory
- Verification: Use our calculator to check your work