Differentiability Study Guide- Concepts and Theorems
What Is Differentiability?
Differentiability answers one question: does a function have a tangent line at a given point? If the derivative exists, the function is differentiable there. If not, you've got problems.
A function f(x) is differentiable at x = a if this limit exists:
f'(a) = lim(h→0) [f(a+h) - f(a)] / h
That's it. One limit. If it exists, you're done. If it doesn't, the function is not differentiable at that point.
Differentiability vs. Continuity
Here's something students mess up constantly: differentiability implies continuity, but continuity does not imply differentiability.
If a function is differentiable at a point, it MUST be continuous there. The reverse is false.
Classic counterexample: f(x) = |x| at x = 0. It's continuous everywhere. But the left-hand derivative is -1, the right-hand derivative is +1. They don't match. No derivative exists. Not differentiable.
When Continuity Fails, Differentiability Fails
A jump discontinuity kills differentiability instantly. The limit from the left and right don't even agree on the function value, so there's no chance of a derivative existing.
Vertical asymptotes are even worse. The function blows up to infinity. No tangent line can exist when the function itself breaks apart.
Theorems You Need to Know
1. Sum/Difference Rule
If f and g are differentiable at a, then f ± g is differentiable at a, and:
(f ± g)'(a) = f'(a) ± g'(a)
No surprises here. Differentiation is linear.
2. Product Rule
For the product of two differentiable functions:
(fg)'(a) = f'(a)g(a) + f(a)g'(a)
Memorize this. You'll use it constantly. Don't try to distribute the derivative through a product—it doesn't work.
3. Quotient Rule
For f/g where g(a) ≠ 0:
(f/g)'(a) = [f'(a)g(a) - f(a)g'(a)] / [g(a)]²
The numerator follows the pattern: derivative of top times bottom, minus top times derivative of bottom. The denominator is always the bottom squared.
4. Chain Rule
This is where students fall apart. If y = f(g(x)) and g is differentiable at a, and f is differentiable at g(a), then:
(f ∘ g)'(a) = f'(g(a)) · g'(a)
You take the derivative of the outer function evaluated at the inner function, then multiply by the derivative of the inner function. Practice this one until it's automatic.
5. The Mean Value Theorem
If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) where:
f'(c) = [f(b) - f(a)] / (b - a)
This theorem guarantees that somewhere between a and b, the instantaneous rate of change equals the average rate of change. It's the backbone of why calculus works.
Where Differentiability Breaks Down
Functions fail to be differentiable at a point when:
- The function has a corner (like |x| at 0)
- The function has a cusp (sharp point where slopes go to ±∞)
- The function has a vertical tangent line
- The function is discontinuous at that point
- The function is not defined at that point
Visualize it this way: if you can't draw a single, clean tangent line that touches the graph at exactly one point and doesn't cross it, the derivative doesn't exist.
One-Sided Differentiability
The derivative is a two-sided limit. Sometimes you need to check left and right separately.
A function is differentiable at a point if and only if both one-sided derivatives exist AND they're equal. If the left-hand derivative exists but the right-hand one doesn't, or if they disagree, you get no derivative.
Higher Order Derivatives
The derivative of a derivative is the second derivative. You keep going:
- f'(x) — rate of change
- f''(x) — rate of change of the rate of change (acceleration)
- f'''(x) — jerk (yes, that's the technical term)
Notationally, you'll see f⁽ⁿ⁾(x) for the nth derivative. The pattern continues indefinitely for "nice" functions.
Comparing Differentiability Scenarios
| Function | At Point | Continuous? | Differentiable? | Why? |
|---|---|---|---|---|
| f(x) = x² | x = 1 | Yes | Yes | Clean derivative, no corners |
| f(x) = |x| | x = 0 | Yes | No | Corner — slopes don't match |
| f(x) = 1/x | x = 0 | No | No | Undefined at point |
| f(x) = ∛x | x = 0 | Yes | No | Vertical tangent |
| f(x) = x^(1/3) | x = 0 | Yes | No | Derivative goes to infinity |
How to Check Differentiability: Step by Step
When you're given a function and asked if it's differentiable at a point, here's what you do:
Step 1: Check if the function is defined
If f(a) doesn't exist, you can stop. It's not differentiable. Move on.
Step 2: Check for continuity
Calculate lim(x→a) f(x). Does it equal f(a)? If no, stop. Not differentiable.
Step 3: Compute the derivative
Use the limit definition or differentiation rules to find f'(x). Plug in x = a.
Step 4: Check if the limit exists
Make sure the derivative formula doesn't produce 0/0 or any undefined behavior at x = a. If it does, go back to the limit definition and evaluate carefully.
Step 5: For piecewise functions
Check each piece. Then check the boundary points where the pieces meet. Corners and jumps hide at these boundaries.
Common Mistakes That Cost You Points
- Assuming continuity means differentiability. It doesn't. |x| is the textbook example.
- Forgetting the chain rule. When you have a function inside another function, you need both derivatives.
- Multiplying derivatives incorrectly. (fg)' is NOT f'g'. That's the product rule, not distribution.
- Ignoring the domain. A function can be differentiable everywhere it's defined and that's fine.
- Skipping the limit definition. When in doubt, go back to first principles.
The Bottom Line
Differentiability comes down to one thing: does the limit defining the derivative exist? Everything else—the theorems, the rules, the examples—flows from that definition.
Know what makes a derivative fail (corners, cusps, discontinuities, vertical tangents). Know your product, quotient, and chain rules. Know that differentiability always requires continuity, but the reverse is a trap.
That's the entire game.