Elementary Derivative Rules

Elementary Derivative Rules

Motivating Questions

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?

2.1.1 Introduction: From Limits to Rules

• 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