2.1 – Derivative Definition and Basic Rules
Learning Objectives
By the end of this section, you should be able to:
- Understand the limit definition of the derivative
- Apply the power rule, constant rule, sum/difference rule, and constant multiple rule
- Differentiate polynomial functions
- Interpret derivatives as instantaneous rates of change
- Solve basic derivative problems using multiple approaches
Definition of the Derivative
The derivative of a function at a point represents the instantaneous rate of change of the function at that point. Geometrically, it represents the slope of the tangent line to the function's graph.
The derivative of function \( f(x) \) is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
This is sometimes called the difference quotient or the limit definition of the derivative.
Example 1: Let \( f(x) = x^2 \). Find the derivative using the limit definition.
Solution:
\[ f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} \]
\[ = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \]
\[ = \lim_{h \to 0} \frac{2xh + h^2}{h} \]
\[ = \lim_{h \to 0} (2x + h) = 2x \]
So, \( f'(x) = 2x \).
Key Concept: Alternative Notation
The derivative can be represented in several ways:
- \( f'(x) \) - Lagrange notation
- \( \frac{dy}{dx} \) - Leibniz notation
- \( \frac{d}{dx}[f(x)] \) - Operator notation
- \( \dot{y} \) - Newton's notation (less common in calculus)
Basic Rules of Differentiation
| Rule Name | Function | Derivative |
|---|---|---|
| Constant Rule | \( f(x) = c \) | \( f'(x) = 0 \) |
| Power Rule | \( f(x) = x^n \) | \( f'(x) = nx^{n-1} \) |
| Constant Multiple Rule | \( f(x) = c \cdot g(x) \) | \( f'(x) = c \cdot g'(x) \) |
| Sum/Difference Rule | \( f(x) = g(x) \pm h(x) \) | \( f'(x) = g'(x) \pm h'(x) \) |
Worked Examples
Example 1: Quadratic Function
Power Rule ApplicationProblem: Find the derivative of \( f(x) = x^2 + 2x - 5 \) and evaluate it at \( x = 3 \).
Apply the power rule to each term:
\[ f(x) = x^2 + 2x - 5 \]
\[ f'(x) = \frac{d}{dx}[x^2] + \frac{d}{dx}[2x] - \frac{d}{dx}[5] \]
Differentiate each term:
\[ \frac{d}{dx}[x^2] = 2x \]
\[ \frac{d}{dx}[2x] = 2 \]
\[ \frac{d}{dx}[5] = 0 \]
Combine the results:
\[ f'(x) = 2x + 2 \]
Evaluate at \( x = 3 \):
\[ f'(3) = 2(3) + 2 = 6 + 2 = 8 \]
Interpretation: At \( x = 3 \), the function \( f(x) = x^2 + 2x - 5 \) has an instantaneous rate of change of 8. This means the slope of the tangent line at that point is 8.
Example 2: Function with Radical (Surd Form)
Power Rule with Fractional ExponentsProblem: Find the derivative of \( g(x) = 3\sqrt{x} + \frac{2}{\sqrt{x}} \).
Rewrite using fractional exponents:
\[ g(x) = 3x^{1/2} + 2x^{-1/2} \]
Apply the power rule to each term:
\[ g'(x) = \frac{d}{dx}[3x^{1/2}] + \frac{d}{dx}[2x^{-1/2}] \]
Differentiate each term:
\[ \frac{d}{dx}[3x^{1/2}] = 3 \cdot \frac{1}{2}x^{-1/2} = \frac{3}{2}x^{-1/2} \]
\[ \frac{d}{dx}[2x^{-1/2}] = 2 \cdot \left(-\frac{1}{2}\right)x^{-3/2} = -x^{-3/2} \]
Combine the results:
\[ g'(x) = \frac{3}{2}x^{-1/2} - x^{-3/2} \]
Rewrite in radical form:
\[ g'(x) = \frac{3}{2\sqrt{x}} - \frac{1}{x\sqrt{x}} \]
\[ g'(x) = \frac{3}{2\sqrt{x}} - \frac{1}{x^{3/2}} \]
Key Insight: When working with radicals, convert to fractional exponents before differentiating. Remember that \( \sqrt{x} = x^{1/2} \) and \( \frac{1}{\sqrt{x}} = x^{-1/2} \).
Example 3: Third-Degree Polynomial
Combining Multiple RulesProblem: Find the derivative of \( h(x) = 2x^3 - 5x^2 + 3x - 7 \).
Apply differentiation rules term by term:
\[ h(x) = 2x^3 - 5x^2 + 3x - 7 \]
\[ h'(x) = \frac{d}{dx}[2x^3] - \frac{d}{dx}[5x^2] + \frac{d}{dx}[3x] - \frac{d}{dx}[7] \]
Apply the constant multiple and power rules:
\[ \frac{d}{dx}[2x^3] = 2 \cdot 3x^{2} = 6x^2 \]
\[ \frac{d}{dx}[5x^2] = 5 \cdot 2x^{1} = 10x \]
\[ \frac{d}{dx}[3x] = 3 \cdot 1x^{0} = 3 \]
\[ \frac{d}{dx}[7] = 0 \]
Combine the results:
\[ h'(x) = 6x^2 - 10x + 3 \]
Interpret the derivative:
The derivative \( h'(x) = 6x^2 - 10x + 3 \) represents the instantaneous rate of change of the cubic function \( h(x) \). This quadratic function tells us how the slope of \( h(x) \) changes at each point.
Pattern Recognition: For any polynomial, the derivative will be another polynomial of one degree lower. A cubic function (degree 3) has a quadratic derivative (degree 2).
Tip: Always simplify before differentiating, and remember these basic rules—they form the foundation of all derivative calculations.
AP Exam Tip
On the AP Calculus exam, you may be asked to:
- Find derivatives using the limit definition (especially for simple functions)
- Apply basic differentiation rules to polynomial functions
- Interpret the meaning of a derivative in context
- Find the equation of a tangent line at a point
Practice these skills thoroughly as they appear frequently on both the multiple-choice and free-response sections.
Check Yourself
1. Find \( \frac{d}{dx}[3x^4 - 5x + 7] \)
2. Differentiate \( f(x) = 7 \)
3. Find the derivative of \( f(x) = 2x^3 - 4x^2 + x - 8 \)
4. Use the limit definition to find the derivative of \( f(x) = 3x + 2 \)
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
\[ = \lim_{h \to 0} \frac{[3(x+h) + 2] - [3x + 2]}{h} \]
\[ = \lim_{h \to 0} \frac{3x + 3h + 2 - 3x - 2}{h} \]
\[ = \lim_{h \to 0} \frac{3h}{h} = \lim_{h \to 0} 3 = 3 \]
So, \( f'(x) = 3 \).