Derivatives of Elementary Functions

This section covers the derivate fondamentali and techniques for finding derivatives of basic functions.

Key points:

• Derivative of a constant function: f(x) = c → f'(x) = 0
• Derivative of f(x) = x: f'(x) = 1
• Power rule: If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹
• Derivatives of trigonometric functions: (sin x)' = cos x, (cos x)' = -sin x
• Derivative of exponential function: (eˣ)' = eˣ
• Derivative of logarithmic function: (ln x)' = 1/x

Definition: The power rule states that for any real number n, the derivative of xⁿ is nxⁿ⁻¹

Example: The derivative of f(x) = x³ is f'(x) = 3x²

Highlight: The exponential function eˣ is unique in that it is its own derivative

Algebra of Derivatives and Composite Functions

This section explores rules for combining derivatives and handling composite functions.

Key rules:

• Sum rule: (f + g)' = f' + g'
• Difference rule: (f - g)' = f' - g'
• Product rule: (fg)' = f'g + fg'
• Quotient rule: (f/g)' = (f'g - fg') / g²
• Chain rule for composite functions: (f ∘ g)' = (f' ∘ g) · g'

Example: If y = sin(x²), applying the chain rule gives y' = cos(x²) · 2x

Highlight: The chain rule is crucial for differentiating composite functions and is widely used in calculus applications

Vocabulary: Funzione composta (Composite function) - A function formed by applying one function to the result of another

These rules form the foundation for differentiating complex functions and are essential tools in calculus and its applications across various fields of science and engineering.