Lesson 2: The Derivative — Definition and Notation
Estimated time: 35-45 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- State the formal limit definition of the derivative
- Compute derivatives using the limit definition
- Recognize and use Leibniz, Lagrange, and Newton notations
- Find the derivative function f'(x) from scratch
- Write equations of tangent lines using the derivative
The Formal Definition
Definition of the Derivative
The derivative of f at x = a is: f'(a) = limh→0 [f(a + h) − f(a)] / h
provided this limit exists. We say f is differentiable at a.
The derivative function f'(x) is obtained by leaving a as a variable:
f'(x) = limh→0 [f(x + h) − f(x)] / h
Notation
Common Derivative Notations
All of the following mean "the derivative of y = f(x) with respect to x":
- Lagrange notation: f'(x) ("f prime of x")
- Leibniz notation: dy/dx or df/dx ("dee y dee x")
- Operator notation: Df(x) or Dxf
- At a specific point: f'(a) or dy/dx |x=a
Computing Derivatives from the Definition
Example 1: f(x) = x³
Step 1: f(x+h) = (x+h)³ = x³ + 3x²h + 3xh² + h³
Step 2: f(x+h) − f(x) = 3x²h + 3xh² + h³
Step 3: Divide by h: 3x² + 3xh + h²
Step 4: limh→0 (3x² + 3xh + h²) = 3x²
Therefore f'(x) = 3x².
Example 2: f(x) = √x
Step 1: [√(x+h) − √x]/h. Rationalize by multiplying by [√(x+h) + √x]/[√(x+h) + √x].
Step 2: Numerator: (x+h) − x = h. Result: h/[h(√(x+h) + √x)] = 1/[√(x+h) + √x].
Step 3: limh→0 = 1/(2√x). Therefore f'(x) = 1/(2√x).
Example 3: f(x) = 1/x
Step 1: [1/(x+h) − 1/x]/h = [x − (x+h)]/[hx(x+h)] = −h/[hx(x+h)]
Step 2: Cancel h: −1/[x(x+h)]
Step 3: limh→0 = −1/x². Therefore f'(x) = −1/x².
The Derivative as a Function
The derivative f'(x) is itself a function. Its domain is the set of all x where the limit definition produces a finite value. For each x in the domain, f'(x) gives the slope of the tangent line at that point.
Example 4: Tangent Line Equation
Find the tangent line to f(x) = x³ at x = 2.
Step 1: f(2) = 8. f'(x) = 3x², so f'(2) = 12.
Step 2: Tangent line: y − 8 = 12(x − 2), or y = 12x − 16.
Alternative Form of the Derivative
Alternative Definition
f'(a) = limx→a [f(x) − f(a)] / (x − a)
This form is sometimes more convenient for specific computations.
Interpreting the Derivative
The derivative f'(a) has multiple interpretations:
- Geometric: Slope of the tangent line to y = f(x) at x = a
- Physical: Instantaneous rate of change of f with respect to x at x = a
- Kinematic: If s(t) is position, then s'(t) is velocity; v'(t) is acceleration
Key Takeaways
- The derivative f'(x) = limh→0 [f(x+h) − f(x)]/h is a function giving slopes of tangent lines.
- Common notations: f'(x), dy/dx, Df(x) all mean the same thing.
- To compute from the definition: expand f(x+h), subtract f(x), divide by h, take limit.
- The derivative of xn is nxn−1 (we will prove this with the power rule soon).
Check Your Understanding
1. Use the limit definition to find f'(x) for f(x) = 5x − 2.
2. Find the tangent line to f(x) = 1/x at x = 1.
3. What does f'(5) = 3 tell you geometrically?