Page 1: Fundamentals of Exponential Functions and Derivative Rules

This page introduces the core concepts of exponential functions and their derivatives, with particular emphasis on complex function differentiation. The content explores the product rule and chain rule applications in exponential contexts.

Definition: The exponential function with base e (Euler's number) is a fundamental mathematical function where e ≈ 2.718.

Highlight: The exponential function never crosses the x-axis but approaches it asymptotically, meaning it has no real zeros.

Example: For f(x) = 3x³(x+1), the derivative is calculated using the product rule: f'(x) = 9x²(x+1) + 3x³·1

Vocabulary: The product rule states that for f(x) = u·v, the derivative is f'(x) = u'v + uv'

Key properties of exponential functions are thoroughly explained, including:

• The limit behavior as x approaches negative infinity: lim(e^x) = 0
• The limit behavior as x approaches positive infinity: lim(e^x) = ∞