Introduction to Derivatives
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Average Rate of Change
Definition:
The average rate of change measures how much a function changes on average over an interval
\[ \text{Average Rate of Change} = \frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a} \]
where:
• [a, b] is the interval
• f(a) and f(b) are function values at endpoints
• This is the slope of the secant line through (a, f(a)) and (b, f(b))
Geometric Interpretation:
The average rate of change is the slope of the secant line connecting two points on the curve
• Secant line: A line passing through two points on a curve
• Represents average behavior over an interval
Example:
Find average rate of change of \( f(x) = x^2 \) on [1, 3]
a = 1, b = 3
f(1) = 1² = 1
f(3) = 3² = 9
Average rate = \( \frac{9 - 1}{3 - 1} = \frac{8}{2} = 4 \)
The function increases by 4 units per unit change in x
2. Instantaneous Rate of Change
Definition:
The instantaneous rate of change is the rate at which a function changes at a specific point (not over an interval)
\[ \text{Instantaneous Rate} = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]
\[ \text{Or: } \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]
• This is the limit of average rates as the interval shrinks to zero
• Represents the exact rate at a single point
• This limit is called the derivative
From Average to Instantaneous:
As the second point gets closer to the first point (h → 0), the secant line approaches the tangent line
• Average rate → Instantaneous rate
• Secant line → Tangent line
3. The Derivative
Definition Using Limits:
The derivative of f(x) at x = a is:
\[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]
\[ \text{Or: } f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]
Notation:
• \( f'(a) \) or \( \frac{df}{dx}\bigg|_{x=a} \) or \( \frac{dy}{dx}\bigg|_{x=a} \)
• Read as: "f prime of a" or "the derivative of f at a"
What Does the Derivative Tell Us?
• Instantaneous rate of change at that point
• Slope of tangent line to the curve at that point
• How fast the function is changing at that exact moment
Example:
Find \( f'(2) \) for \( f(x) = x^2 \) using the limit definition
\( f'(2) = \lim_{h \to 0} \frac{f(2+h) - f(2)}{h} \)
\( = \lim_{h \to 0} \frac{(2+h)^2 - 4}{h} \)
\( = \lim_{h \to 0} \frac{4 + 4h + h^2 - 4}{h} \)
\( = \lim_{h \to 0} \frac{4h + h^2}{h} \)
\( = \lim_{h \to 0} (4 + h) \)
\( = 4 \)
4. Velocity as a Rate of Change
Position, Velocity, and Derivatives:
If s(t) represents position at time t:
Average Velocity:
\[ v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} = \frac{\Delta s}{\Delta t} \]
Instantaneous Velocity:
\[ v(t) = s'(t) = \lim_{h \to 0} \frac{s(t+h) - s(t)}{h} \]
Interpretation:
• Positive velocity: Moving forward/upward
• Negative velocity: Moving backward/downward
• Zero velocity: Momentarily at rest
• Speed = |velocity| (magnitude only)
Example:
Position: \( s(t) = -16t^2 + 64t \) (feet), find velocity at t = 2 seconds
\( v(2) = \lim_{h \to 0} \frac{s(2+h) - s(2)}{h} \)
s(2) = -16(4) + 64(2) = 64 feet
After limit calculation (similar to previous example):
v(2) = 0 feet/second (object is momentarily at rest)
5. Slope of Tangent Line
Definition:
The tangent line to a curve at a point is the line that just "touches" the curve at that point
\[ m_{tangent} = f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \]
Key Point:
The slope of the tangent line equals the derivative at that point
Finding the Slope:
1. Set up the limit definition with the given point
2. Expand and simplify the numerator
3. Factor and cancel h (or x - a)
4. Evaluate the limit as h → 0
Example:
Find slope of tangent to \( f(x) = x^2 + 1 \) at x = 3
\( m = \lim_{h \to 0} \frac{[(3+h)^2 + 1] - [9 + 1]}{h} \)
\( = \lim_{h \to 0} \frac{9 + 6h + h^2 + 1 - 10}{h} \)
\( = \lim_{h \to 0} \frac{6h + h^2}{h} \)
\( = \lim_{h \to 0} (6 + h) \)
Slope = 6
6. Equation of Tangent Line
Point-Slope Form:
Once you have the slope and a point, use point-slope form:
\[ y - f(a) = f'(a)(x - a) \]
where:
• (a, f(a)) is the point of tangency
• f'(a) is the slope (derivative at x = a)
Steps to Find Tangent Line Equation:
1. Find the point: Calculate f(a) to get (a, f(a))
2. Find the slope: Calculate f'(a) using the limit definition
3. Use point-slope form: Plug values into \( y - y_1 = m(x - x_1) \)
4. Simplify: Write in slope-intercept form if needed
Complete Example:
Find equation of tangent to \( f(x) = x^2 \) at x = 2
Step 1: Find the point
f(2) = 4, so point is (2, 4)
Step 2: Find the slope
From earlier example: f'(2) = 4
Step 3: Use point-slope form
y - 4 = 4(x - 2)
Step 4: Simplify
y - 4 = 4x - 8
y = 4x - 4
7. Techniques for Finding Derivatives Using Limits
Common Strategies:
1. Algebraic Simplification:
Expand, combine like terms, factor out h
2. Factoring:
Factor numerator and cancel common factor with denominator
3. Rationalization:
Multiply by conjugate when square roots are involved
4. Common Denominator:
Combine fractions before taking the limit
8. Quick Reference Summary
Key Formulas:
Average Rate: \( \frac{f(b) - f(a)}{b - a} \)
Instantaneous Rate (Derivative): \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \)
Tangent Slope: \( m = f'(a) \)
Tangent Equation: \( y - f(a) = f'(a)(x - a) \)
Velocity: \( v(t) = s'(t) \)
📚 Study Tips
✓ Average rate = slope of secant line (over interval)
✓ Instantaneous rate = derivative = slope of tangent line (at a point)
✓ Derivative is the limit of average rates as interval shrinks to zero
✓ Always factor and cancel h before evaluating the limit
✓ Tangent line touches curve at exactly one point (locally)