Page 2: Composite Functions and Advanced Operations

This page delves into the composition of functions and their derivatives, presenting more complex derivative operations and special cases.

Definition: For composite functions, if u is derivable on I and v is derivable on u(I), then v(u(x)) is derivable on I with the derivative (v(u(x)))' = u'(x) × v'(u(x)).

Example: Special cases include:

1. Reciprocal function: (1/u(x))' = -u'(x)/u²(x)
2. Square root function: (√u(x))' = u'(x)/(2√u(x))

Highlight: The quotient rule for two functions states that if v doesn't vanish on I, then (u/v)' = (u'v - uv')/v².

Page 1: Fundamental Derivative Concepts

This page introduces the core concepts of derivatives and basic derivative operations. The content establishes the foundation for understanding function derivatives and their calculations.

Definition: A function f is considered "derivable" on an interval I when it admits a derivative number for every point in that interval.

Vocabulary: The derivative function, denoted as f', maps each point x to its derivative value f'(x).

Example: Basic derivative functions include:

• Constant function f(x) = a → f'(x) = 0
• Power function f(x) = xⁿ → f'(x) = nx^(n-1)

Highlight: Three fundamental operations on derivatives are presented:

1. Sum rule: (u + v)' = u' + v'
2. Constant multiple rule: (ku)' = k × u'
3. Product rule: (u × v)' = u'v + uv'