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Chain Rule for Derivatives

Sep 10, 2025

Overview

This lecture centers on the chain rule for derivatives, covering its purpose, a sketch of its proof, how to apply it, and how it compares to other derivative rules. The session includes detailed examples, practice problems, and abstract applications involving function composition and tangent lines.

Derivative Rules Overview

  • The course has introduced rules for finding derivatives of common functions:
    • Power functions: x^k
    • Exponential: e^x
    • Trigonometric: sin(x), cos(x), tan(x), sec(x)
  • More functions exist beyond these, often formed by combining basic functions.
  • Key rules discussed so far:
    • Constant multiple rule: The derivative of a constant times a function is the constant times the derivative.
    • Sum (additive) rule: The derivative of a sum is the sum of the derivatives.
    • Product rule: For two functions f and g, (fg)' = f'g + fg'.
    • Quotient rule: For f/g, the derivative is (f'g - fg')/g^2.

The Chain Rule

  • The chain rule is used when differentiating a composition of functions (a function inside another function).
  • For y = f(g(x)), the derivative is f'(g(x)) × g'(x).
  • Steps:
    1. Differentiate the outside function, leaving the inside unchanged.
    2. Multiply by the derivative of the inside function.
  • Notation: f(g(x)) or (f ∘ g)(x) highlights the composition.
  • The chain rule is essential for handling complex, layered functions and is foundational for advanced calculus topics.

Sketch of the Proof

  • The proof uses the limit definition of the derivative for f(g(x)):
    • limₕ→0 [f(g(x+h)) - f(g(x))]/h
  • To facilitate the chain rule, multiply by 1 in the form [g(x+h) - g(x)] / [g(x+h) - g(x)].
  • This allows the expression to be split and interpreted as the product of two derivatives:
    • One part becomes f' evaluated at g(x).
    • The other part becomes g'(x).
  • The proof is a sketch, omitting technical details, but provides intuition for why the chain rule works.

Chain Rule Examples

  • d/dx [e^(2x)] = e^(2x) × 2
  • d/dx [e^(-x)] = e^(-x) × (-1)
  • d/dx [sin(2x)] = cos(2x) × 2
  • d/dx [tan(½x)] = sec^2(½x) × ½
  • These examples show how the chain rule simplifies the process compared to earlier, more laborious methods.

Applying the Chain Rule

  • For complex, layered functions ("onion problems"), apply the chain rule repeatedly, working from the outermost function inward.
  • Example:
    • d/dx [tan(√(e^(x^2+5) + sin(x^2)))]
      • Rewrite roots as exponents for easier differentiation.
      • Apply the chain rule at each layer: tangent, then square root, then sum, then exponentials and trigonometric functions.
      • Carefully multiply by the derivative of each inner function as you work inward.
  • The key is to proceed step by step, applying the appropriate rule at each layer.

Comparing Methods: Expansion vs. Chain Rule

  • For expressions like (x^2 + 3x)^2:
    • Expanding first and then differentiating yields the same result as using the chain rule.
  • For higher powers, such as (x^2 + 3x)^100:
    • Expansion is impractical.
    • The chain rule provides an efficient method:
      • d/dx [(x^2 + 3x)^100] = 100(x^2 + 3x)^99 × (2x + 3)
  • The chain rule is especially valuable for large exponents or complicated compositions.

Abstract Example: Function Composition and Tangents

  • The tangent line to y = h(x) at x = a encodes two key pieces of information:
    • h(a): the value of the function at a
    • h'(a): the derivative (slope) at a
  • Given a tangent line to y = f(x) at x = 2, you can find:
    • f(2) by plugging x = 2 into the tangent line equation.
    • f'(2) as the slope of the tangent line.
  • For composite functions like f(x^2−x), (f(x))^2, or f(f(x)), use the chain rule and the known values of f and f' at the relevant points to find the tangent lines at x = 2.
  • The process involves:
    • Calculating the function value at the point.
    • Using the chain rule to find the derivative at that point.
    • Writing the tangent line equation using these values.

Table Practice: Evaluating Chain Rule From Values

  • For h(x) = f(g(x)), the derivative is h'(x) = f'(g(x)) × g'(x).
  • For k(x) = g(f(x)), the derivative is k'(x) = g'(f(x)) × f'(x).
  • To evaluate derivatives at specific points:
    • Plug the point into the composition.
    • Use the table to find the necessary function and derivative values.
    • Multiply as required by the chain rule.
  • The order of composition matters; f(g(x)) and g(f(x)) are not interchangeable.

Key Terms & Definitions

  • Chain Rule: A rule for differentiating compositions of functions: d/dx [f(g(x))] = f'(g(x)) × g'(x).
  • Inside Function: The inner function in a composition, g(x) in f(g(x)).
  • Outside Function: The outer function in a composition, f in f(g(x)).
  • Onion Problem: A problem with multiple nested functions, requiring repeated application of the chain rule.
  • Tangent Line: A line that touches a curve at a point, matching both the value and the slope of the function at that point.

Action Items / Next Steps

  • Practice differentiating composite functions using the chain rule, including multi-layered ("onion") problems.
  • Review problems involving tables of values and layered compositions to reinforce understanding.
  • Prepare for upcoming problems that may involve interpreting data from tables and graphs, and applying the chain rule in various formats.

Full transcript