3.6 The Chain Rule - Calculus Volume 1

Learning Objectives

• 3.6.1 State the chain rule for the composition of two functions.
• 3.6.2 Apply the chain rule together with the power rule.
• 3.6.3 Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
• 3.6.4 Recognize the chain rule for a composition of three or more functions.
• 3.6.5 Describe the proof of the chain rule.

Introduction to the Chain Rule

• Techniques already learned include differentiating basic functions, sums, differences, products, quotients, and constant multiples.
• These techniques do not cover differentiating compositions of functions such as:
  • $h(x) = f(g(x))$

Deriving the Chain Rule

• Concept: For a function that is a composition, the derivative is not simply the sum of derivatives.
• Chain Rule: The derivative of a composite function is given by:
  $(f(g(x)))' = f'(g(x)) imes g'(x)$
• Contextual Example: If we consider how one variable's change affects another in the chain:
  • If $x$ changes, $g(x)$ changes, leading to a subsequent change in $f(g(x))$.
  • Recognize that the derivative of $g(x)$ contributes to the final derivative as well.

Formal Derivative Expression

• Set up limit for derivative:
  $h'(x) = \frac{h(x + ext{h}) - h(x)}{ ext{h}}$
• Enhance by multiplying and dividing by $g$ to derive:
  • Take limit as $x$ approaches a specific point, leading to:
    $ext{Example limit setup: } ext{as } x o a: h'(x) = \frac{ ext{[evaluated expression]}}{ ext{a}}$
  • This concept illustrates that the first term in the product is the derivative of the outside function:
• Acknowledgment of using trigonometric derivatives as examples (e.g., $ext{sin}(x)$)

General Case of the Chain Rule

• The Chain Rule: Let $f$ and $g$ be functions where:
  • For all $x$ in the domain of $g$ for which both $f$ and $g$ are differentiable:
    $(f(g(x)))' = f'(g(x)) imes g'(x)$

Applying the Chain Rule

Steps for Differentiating Using the Chain Rule:

1. Identify the outer function, $f$, and the inner function, $g$.
2. Differentiate the outer function $f$, then evaluate it at the inner function $g(x)$.
3. Differentiate the inner function, $g$.
4. Multiply the derivatives obtained:
   $(f(g(x)))' = f'(g(x)) imes g'(x)$

Chain Rule and Power Rule Combined

• Application where functions take form: $f(g(x)) = (g(x))^n$
  • Utilizing Power Rule results in:
    $$ h