3.6 The Chain Rule

The chain rule is a powerful tool for finding derivatives of composite functions. It allows us to break down complex functions into simpler parts, making differentiation easier. This rule is essential for tackling a wide range of mathematical problems in calculus.

Combining the chain rule with other differentiation techniques expands our problem-solving toolkit. From power and product rules to multiple compositions, the chain rule's versatility shines through. Its real-world applications in physics, economics, and engineering make it a crucial concept to master.

The Chain Rule

Chain rule for composite functions

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• Finds derivative of composite function $f(g(x))$ (function composition)
• Identifies outer function $f$ and inner function $g$
• Multiplies derivative of outer function $f'(g(x))$ by derivative of inner function $g'(x)$
• Final derivative: $h'(x) = f'(g(x)) \cdot g'(x)$
• Examples:
  • $h(x) = \sin(x^2)$, outer function $f(x) = \sin(x)$, inner function $g(x) = x^2$
  • $h(x) = e^{\cos(x)}$, outer function $f(x) = e^x$, inner function $g(x) = \cos(x)$
• Calculates the rate of change of nested functions

Combining chain rule with other rules

• Power rule: $h(x) = (g(x))^n$, $h'(x) = n(g(x))^{n-1} \cdot g'(x)$
• Product rule: $h(x) = f(x) \cdot g(x)$, $h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$
  • Apply chain rule if $f(x)$ or $g(x)$ is composite function
• Quotient rule: $h(x) = \frac{f(x)}{g(x)}$, $h'(x) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}$
  • Apply chain rule if $f(x)$ or $g(x)$ is composite function
• Examples:
  • $h(x) = (x^2 + 1)^3$, power rule and chain rule
  • $h(x) = \sin(x) \cdot e^x$, product rule and chain rule

Chain rule for multiple compositions

• Applies to compositions of three or more functions $h(x) = f(g(k(x)))$
• Derivative: $h'(x) = f'(g(k(x))) \cdot g'(k(x)) \cdot k'(x)$
• Work from outside in, applying chain rule at each step
• Multiply derivatives of each function in composition
• Example: $h(x) = \ln(\sin(e^x))$
  1. Outer function $f(x) = \ln(x)$, middle function $g(x) = \sin(x)$, inner function $k(x) = e^x$
  2. $h'(x) = \frac{1}{\sin(e^x)} \cdot \cos(e^x) \cdot e^x$

Mathematical basis of chain rule

• Based on concept of composite function with "inner" and "outer" functions
• Chain rule calculates how changes in $x$ affect $g(x)$ and then how changes in $g(x)$ affect $f(g(x))$
• Justified using limit definition of derivative
  • Applying limit definition to composite function shows derivative is product of outer and inner function derivatives

Real-world applications of chain rule

• Useful for solving problems involving rates of change in physics, economics, engineering
• Velocity and acceleration:
  • Position $s(t)$, velocity $v(t) = s'(t)$, acceleration $a(t) = v'(t)$
  • Use chain rule if $s(t)$ is composite function
• Marginal cost and revenue in economics:
  • Cost $C(x)$, revenue $R(x)$, use chain rule if $C(x)$ or $R(x)$ are composite functions
• Optimization problems in engineering:
  • Objective function or constraints involve composite functions
  • Chain rule helps find optimal solution by calculating necessary derivatives

Variables and Implicit Differentiation

• Dependent variable: The output of a function, typically y or f(x)
• Independent variable: The input of a function, typically x
• Implicit differentiation: A technique using the chain rule to find derivatives of implicitly defined functions
  • Useful when a function is not explicitly solved for the dependent variable