Calculus: Composite Functions and Derivatives

Composite Functions

• Definition: A function composed of multiple interrelated functions.
• Example: h(x) = cos(x^2).
  • f(x) = cos(x) is the outer function.
  • g(x) = x^2 is the inner function.
• Chain Rule: Used to differentiate composite functions.
  • Formula: If h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).
• Multi-Layered Functions: The chain rule can be repeatedly applied to functions with multiple layers.

Inverse Differentiation

• Definition: A method for finding the derivative of an inverse function.
• Formula: f'(f⁻¹(x)) = 1 / f'(f⁻¹(x)) (Corrected to use correct notation)
• Application: Useful when provided tables of function values and derivative values.

Implicit Differentiation

• Definition: A technique for differentiating functions that aren't explicitly solved for y in terms of x.
• Key Notation: dy/dx represents the derivative of y with respect to x.
• Chain Rule Application: The chain rule applies when differentiating terms containing y. For example, the derivative of y² is 2y * (dy/dx).
• Steps:
  • Step 1: Differentiate both sides of the equation with respect to x.
  • Step 2: Collect all dy/dx terms on one side of the equation.
  • Step 3: Factor out dy/dx from the terms on the side where it appears.
  • Step 4: Isolate dy/dx by dividing both sides of the equation by the factored expression.
• Second Derivative: Calculating the second derivative involves differentiating the result of implicit differentiation, resulting in expressions containing dy/dx.
  • Key Notation: d²y/dx² represents the second derivative.
  • Substitution: The dy/dx expression is often substituted back into the second derivative equation for simplification.

This quiz explores essential concepts in calculus, including composite functions, inverse differentiation, and implicit differentiation. You'll be tested on definitions, formulas, and applications of these techniques, which are critical for understanding advanced calculus concepts.