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Differentiable

Differentiable

A curve that is smooth and contains no discontinuities or cusps. Formally, a curve is differentiable at all values of the domain variable(s) for which the derivative exists.

 

 

See also

Continuous, continuously differentiable, Rolle’s theorem, mean value theorem

Key Formula

f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
Where:
  • ff = The function being tested for differentiability
  • aa = The specific point where you are checking whether the derivative exists
  • hh = A small increment approaching zero

Worked Example

Problem: Determine whether f(x) = |x| is differentiable at x = 0.
Step 1: Write the definition of the derivative at x = 0 using the limit.
f′(0)=lim⁡h→0f(0+h)−f(0)h=lim⁡h→0∣h∣−0h=lim⁡h→0∣h∣hf'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} = \lim_{h \to 0} \frac{|h| - 0}{h} = \lim_{h \to 0} \frac{|h|}{h}
Step 2: Check the limit from the right (h > 0). When h is positive, |h| = h.
lim⁡h→0+hh=1\lim_{h \to 0^+} \frac{h}{h} = 1
Step 3: Check the limit from the left (h < 0). When h is negative, |h| = −h.
lim⁡h→0−−hh=−1\lim_{h \to 0^-} \frac{-h}{h} = -1
Step 4: Compare the one-sided limits. Since 1 ≠ −1, the two-sided limit does not exist.
lim⁡h→0∣h∣h does not exist\lim_{h \to 0} \frac{|h|}{h} \text{ does not exist}
Answer: f(x) = |x| is NOT differentiable at x = 0 because the left-hand and right-hand limits of the difference quotient are not equal. The graph has a sharp corner (a "V" shape) at the origin.

Another Example

Problem: Show that g(x) = x² is differentiable at x = 3.
Step 1: Apply the limit definition of the derivative at x = 3.
g′(3)=lim⁡h→0(3+h)2−32hg'(3) = \lim_{h \to 0} \frac{(3+h)^2 - 3^2}{h}
Step 2: Expand the numerator and simplify.
=lim⁡h→09+6h+h2−9h=lim⁡h→06h+h2h= \lim_{h \to 0} \frac{9 + 6h + h^2 - 9}{h} = \lim_{h \to 0} \frac{6h + h^2}{h}
Step 3: Factor out h and cancel.
=lim⁡h→0(6+h)=6= \lim_{h \to 0} (6 + h) = 6
Answer: The limit exists and equals 6, so g(x) = x² is differentiable at x = 3, with g'(3) = 6.

Frequently Asked Questions

If a function is differentiable, is it always continuous?
Yes. Differentiability at a point guarantees continuity at that point. If a function is differentiable at x = a, then f must be continuous at x = a. However, the reverse is not true — a function can be continuous at a point without being differentiable there (for example, f(x) = |x| at x = 0).
What are the common reasons a function fails to be differentiable?
A function is not differentiable at a point if the graph has a sharp corner (like |x| at 0), a cusp (like x^(2/3) at 0), a vertical tangent line (like the cube root of x at 0), or a discontinuity (a jump or hole). In each case, the limit definition of the derivative fails to produce a finite value.

Differentiable vs. Continuous

A continuous function has no breaks, jumps, or holes — you can draw it without lifting your pen. A differentiable function is continuous AND smooth, with no sharp corners or cusps. Differentiability is a stronger condition: every differentiable function is continuous, but not every continuous function is differentiable. The classic example is f(x) = |x|, which is continuous everywhere but not differentiable at x = 0.

Why It Matters

Differentiability is the foundation of calculus. When a function is differentiable, you can find its instantaneous rate of change, build tangent-line approximations, and apply powerful theorems like the Mean Value Theorem and Rolle's Theorem. Many real-world models in physics, engineering, and economics assume differentiability so that optimization techniques (finding maxima and minima) work reliably.

Common Mistakes

Mistake: Assuming that a continuous function must be differentiable.
Correction: Continuity does not guarantee differentiability. A function can be continuous at a point yet have a sharp corner or cusp there, making it non-differentiable. Always check the derivative limit separately.
Mistake: Checking only one side of the derivative limit and concluding the function is differentiable.
Correction: The derivative exists at a point only if both the left-hand and right-hand limits of the difference quotient exist AND are equal. You must check both sides.

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