Lecture on Differentiation Rules

Introduction to Derivative Rules

• Objective: Speed up the process of finding derivatives.
• Foundation: We established the conceptual understanding of derivatives in previous sections (3.1 and 3.2).
• Goal: Acquire a set of rules to efficiently find derivatives.

Derivative of Constant and Linear Functions

• Constant Function:
  • Form: $f(x) = c$, where $c$ is a real number.
  • Derivative: $0$ (since the function does not change).
• Linear Function:
  • Form: $f(x) = mx + b$.
  • Derivative: $m$ (the slope of the line).

Power Rule

• For Power Functions:
  • Form: $f(x) = x^n$.
  • Derivative: $f'(x) = nx^{n-1}$.
  • Example: For $x^{1000}$, derivative is $1000x^{999}$.

Linearity of Derivatives

• Linear Operator: Differentiation can be applied term-by-term.
• Expression: $s f(x) + t g(x)$.
  • Derivative: $s f'(x) + t g'(x)$.

Applying the Power Rule

• Example 1: $f(x) = 2x^2 + 3x$
  • Derivative: $f'(x) = 4x + 3$.
• Example 2: $f(x) = x^5 + x^4 + x^3 + x^2 + x + 1$
  • Derivative: $5x^4 + 4x^3 + 3x^2 + 2x + 1$.

Negative and Fractional Power Rules

• Generalization: Power rule applies to negative and fractional powers.
• Examples:
  • $y = x^{-1} - x^{-2} + 2x^{-3} - 3x^{-4}$
  • Derivative: $-x^{-2} + 2x^{-3} - 6x^{-4} + 12x^{-5}$

Simplifying Quotients

• Example: $f(x) = \frac{x^3 - 3x^2 - 2x}{x^2}$
  • Simplified: Terms can be separated and simplified before applying the power rule.

Horizontal Tangents

• Definition: Occur when the slope (derivative) is zero.
• Example: For $f(t) = t^3 - 27t + 5$, find points where $f'(t) = 0$.
  • Solution: Solve $3t^2 - 27 = 0$, find $t = 3, -3$.

Exponential Function Derivatives

• Rule for $e^x$: Derivative of $e^x$ is $e^x$.
• Caution: Do not confuse with power rule.

Higher-Order Derivatives

• Concept: Derivatives can be taken multiple times.
• Notation:
  • Second Derivative: $f''$ or $\frac{d^2y}{dx^2}$.
  • Third Derivative: $f'''$ or $\frac{d^3y}{dx^3}$.
  • Higher Orders: $f^{(n)}$ or $\frac{d^ny}{dx^n}$.
• Examples:
  • For $f(x) = x^5 - 3x^3$, find $f'$, $f''$, $f'''$.
  • For $f(x) = 2e^x$, every derivative remains $2e^x$.