Overview

The derivative measures how a function changes — it's the slope of a curve at any point. This topic introduces what a derivative is, how it’s defined, and the fundamental rules that allow you to compute derivatives of common functions.

Key Concepts and Structures

• Definition of Derivative: f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
• Notation: f'(x), dy/dx, D[f(x)]
• Power Rule: \frac{d}{dx}(x^n) = nx^{n-1}
• Constant Rule: \frac{d}{dx}(c) = 0
• Constant Multiple Rule: \frac{d}{dx}(c \cdot f(x)) = c \cdot f'(x)
• Sum and Difference Rule: \frac{d}{dx}(f \pm g) = f' \pm g'
• Basic Derivatives:
  • \frac{d}{dx}(x^n) = nx^{n-1}
  • \frac{d}{dx}(\sin x) = \cos x; \frac{d}{dx}(\cos x) = -\sin x
  • \frac{d}{dx}(e^x) = e^x; \frac{d}{dx}(\ln x) = 1/x

Quick Tip

Practice Problems

1. Find the derivative of f(x) = 5x^3 - 2x + 7
   Show Solution

   Step 1: Use the power rule: nx^{n-1}

   • \frac{d}{dx}(5x^3) = 15x^2
   • \frac{d}{dx}(-2x) = -2
   • \frac{d}{dx}(7) = 0

   Final Answer: f'(x) = 15x^2 - 2

2. Use the limit definition to find the derivative of f(x) = x^2
   Show Solution

   Step 1: Use the formula: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

   Step 2: Compute f(x+h) = (x + h)^2 = x^2 + 2xh + h^2

   Step 3: f(x+h) - f(x) = x^2 + 2xh + h^2 - x^2 = 2xh + h^2

   Step 4: Divide by h: \frac{2xh + h^2}{h} = 2x + h

   Step 5: Take the limit as h → 0: 2x

   Final Answer: f'(x) = 2x