Lesson 3: Differentiability and Local Linearity
Estimated time: 25-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Determine where a function is differentiable
- Identify the three common ways differentiability fails
- Explain the relationship between differentiability and continuity
- Describe local linearity and its connection to the derivative
When Is a Function Differentiable?
A function f is differentiable at x = a if f'(a) = limh→0 [f(a+h) − f(a)]/h exists. This means the limit must exist and be a finite number.
Differentiability Implies Continuity
If f is differentiable at x = a, then f is continuous at x = a. However, the converse is not true: a function can be continuous but not differentiable.
Where Differentiability Fails
There are three common scenarios where a continuous function fails to be differentiable:
1. Corner or Cusp
The graph has a sharp point where the slope changes abruptly. The left-hand derivative and right-hand derivative exist but differ.
Classic example: f(x) = |x| at x = 0. Left derivative = −1, right derivative = +1.
2. Vertical Tangent
The tangent line is vertical at the point, so the slope is undefined (±∞).
Example: f(x) = x1/3 at x = 0. f'(0) = limh→0 h1/3/h = lim 1/h2/3 = ∞.
3. Discontinuity
If f is not continuous at a, it cannot be differentiable there (since differentiability implies continuity).
Example 1: Absolute Value Function
Show that f(x) = |x| is not differentiable at x = 0.
Left derivative: limh→0− |0+h|/h = limh→0− (−h)/h = −1
Right derivative: limh→0+ |0+h|/h = limh→0+ h/h = 1
Since −1 ≠ 1, the derivative does not exist at x = 0.
Example 2: A Piecewise Function
Is g(x) = { x² if x ≤ 1, and 2x − 1 if x > 1 } differentiable at x = 1?
Step 1: Check continuity. Left limit: 1. Right limit: 2(1) − 1 = 1. g(1) = 1. Continuous. ✓
Step 2: Left derivative: d/dx(x²)|x=1 = 2(1) = 2.
Step 3: Right derivative: d/dx(2x − 1)|x=1 = 2.
Both sides agree, so g is differentiable at x = 1 with g'(1) = 2.
Local Linearity
When a function is differentiable at a point, zooming in closely at that point makes the curve look like a straight line. This property is called local linearity.
Local Linearity
If f is differentiable at x = a, then for x near a: f(x) ≈ f(a) + f'(a)(x − a). This is the tangent line approximation (also called linearization). The closer x is to a, the better the approximation.
Example 3: Linear Approximation
Use local linearity to estimate √(4.1).
Step 1: Let f(x) = √x, a = 4. f(4) = 2, f'(x) = 1/(2√x), f'(4) = 1/4.
Step 2: √(4.1) ≈ f(4) + f'(4)(4.1 − 4) = 2 + (1/4)(0.1) = 2 + 0.025 = 2.025.
(Actual value: √4.1 ≈ 2.02485... Very close!)
Differentiability on Intervals
f is differentiable on an open interval (a, b) if it is differentiable at every point in that interval. For closed intervals [a, b], we use one-sided derivatives at the endpoints.
Key Takeaways
- Differentiable implies continuous, but continuous does NOT imply differentiable.
- Differentiability fails at corners/cusps, vertical tangents, and discontinuities.
- To check differentiability at a junction of a piecewise function: verify continuity AND check that left and right derivatives match.
- Local linearity: a differentiable function looks like its tangent line when you zoom in.
- The tangent line approximation f(x) ≈ f(a) + f'(a)(x − a) is useful for estimation.
Check Your Understanding
1. True or false: Every continuous function is differentiable.
2. Is f(x) = { x² if x ≤ 2, and 4x − 4 if x > 2 } differentiable at x = 2?
3. Use linearization of f(x) = x³ at a = 2 to estimate 2.1³.