Lesson 2: The Chain Rule
Estimated time: 35-45 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Identify composite functions and their inner and outer parts
- State and apply the Chain Rule
- Use the Chain Rule in combination with other rules
- Differentiate functions with multiple layers of composition
Composite Functions Review
A composite function has the form f(g(x)): you first apply g (the inner function), then f (the outer function). For example, (x² + 1)5 is f(g(x)) where g(x) = x² + 1 (inner) and f(u) = u5 (outer).
To differentiate a composite function, you need the Chain Rule.
The Chain Rule
Chain Rule
If y = f(g(x)) and both f and g are differentiable, then:
dy/dx = f'(g(x)) · g'(x)
In words: "the derivative of the outer function evaluated at the inner function, times the derivative of the inner function."
Leibniz Notation
If y = f(u) and u = g(x), then: dy/dx = (dy/du) · (du/dx)
This form makes the Chain Rule look like fraction cancellation (though dy/du and du/dx are not true fractions).
Example 1: Power of a Function
Find d/dx [(x² + 1)5].
Step 1: Outer function: u5. Inner function: u = x² + 1.
Step 2: Derivative of outer: 5u4. Derivative of inner: 2x.
Step 3: Chain Rule: 5(x² + 1)4 · 2x = 10x(x² + 1)4
Example 2: Trig of a Linear Function
Find d/dx [sin(3x)].
Step 1: Outer: sin u, inner: u = 3x.
Step 2: Derivative of outer: cos u. Derivative of inner: 3.
Step 3: Chain Rule: cos(3x) · 3 = 3 cos(3x)
Example 3: Exponential Composition
Find d/dx [ex²].
Step 1: Outer: eu, inner: u = x².
Step 2: Derivative of outer: eu. Derivative of inner: 2x.
Step 3: Chain Rule: ex² · 2x = 2x ex²
The General Power Rule
The Chain Rule combined with the power rule gives us a very useful pattern:
General Power Rule
d/dx [g(x)]n = n [g(x)]n−1 · g'(x)
Example 4: Square Root Composition
Find d/dx [√(3x² + 7)].
Rewrite: (3x² + 7)1/2
Apply General Power Rule: (1/2)(3x² + 7)−1/2 · 6x = 3x / √(3x² + 7)
Chain Rule with Other Rules
Example 5: Chain Rule + Product Rule
Find d/dx [x² sin(3x)].
Product Rule first: d/dx [x²] · sin(3x) + x² · d/dx [sin(3x)]
Chain Rule for sin(3x): d/dx [sin(3x)] = 3 cos(3x)
Combine: 2x sin(3x) + x² · 3 cos(3x) = 2x sin(3x) + 3x² cos(3x)
Example 6: Double Chain Rule
Find d/dx [sin²(4x)] = d/dx [(sin(4x))²].
Outer: u², middle: sin v, inner: v = 4x.
Apply: 2 sin(4x) · cos(4x) · 4 = 8 sin(4x) cos(4x) = 4 sin(8x)
Key Takeaways
- The Chain Rule says: differentiate the outer function, leave the inner function alone, then multiply by the derivative of the inner function.
- In Leibniz notation: dy/dx = (dy/du)(du/dx).
- The General Power Rule is just the Chain Rule applied to [g(x)]n.
- When combining the Chain Rule with other rules, work step by step: identify what rule applies at the top level, then use the Chain Rule for any composite pieces.
Check Your Understanding
1. Find d/dx [(2x − 1)7].
2. Find d/dx [cos(x³)].
3. Find d/dx [esin x].
4. Find d/dx [ln(x² + 5)].