Elementary Derivative Rules

This section addresses fundamental questions regarding derivative computation:

What are the alternate notations used for the derivative?
How can the algebraic structure of a function be leveraged to compute a formula for its derivative, $f'(x)$ ?
What is the derivative of a power function of the form $f(x) = x^n$ ?
What is the derivative of an exponential function of the form $f(x) = a^x$ ?
If the derivative of $y = f(x)$ is known, what is the derivative of $y = kf(x)$ where $k$ is a constant?
If the derivatives of $y = f(x)$ and $y = g(x)$ are known, how is the derivative of $y = f(x) + g(x)$ computed?

Concept of the Derivative: The derivative $f'(x)$ of a function $f$ measures:
- The instantaneous rate of change of $f$ with respect to $x$.
- The slope of the tangent line to the curve $y = f(x)$ at any given value of $x$.
Previous Approach: In Chapter 1, the focus was on interpreting the derivative graphically or as a meaningful rate of change in a physical context. Calculation relied on the limit definition of the derivative:
$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
Current Goal: This chapter aims to derive patterns and rules from the limit definition that allow for rapid computation of derivative formulas, bypassing the direct use of the limit definition.
- Objective: To apply