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Differentiability

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Core Definition

A function f(x)f(x) is differentiable at a point x=ax = a if the derivative f′(a)f'(a) exists:

f′(a)=lim⁡h→0f(a+h)−f(a)hexists and is finite.f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \quad \text{exists and is finite.}
A function is differentiable on an open interval (a,b)(a, b) if it is differentiable at every point in the interval.

Concept Explanation

The Differentiability Implies Continuity Theorem

A fundamental theorem of calculus establishes:

If f(x) is differentiable at x=a, then f(x) is continuous at x=a.\text{If } f(x) \text{ is differentiable at } x = a, \text{ then } f(x) \text{ is continuous at } x = a.

The Converse Is False!

Continuity does NOT guarantee differentiability. A graph can be completely unbroken and continuous, yet fail to possess a derivative at a specific point.

The Four Visual Signatures of Non-Differentiability

A function fails to be differentiable at x=ax = a if any of the following four conditions occur:

  1. Corner / Sharp Turn: The function is continuous, but the one-sided slopes from the left and right do not match:
    lim⁡h→0−f(a+h)−f(a)h≠lim⁡h→0+f(a+h)−f(a)h\lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h} \neq \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}
    Classic example: f(x)=∣x∣f(x) = |x| at x=0x = 0.
  2. Cusp: An extreme sharp point where the curve becomes infinitely steep from both sides with opposite infinite signs (+∞+\infty from one side, −∞-\infty from the other). Classic example: f(x)=x2/3f(x) = x^{2/3} at x=0x = 0.
  3. Vertical Tangent: The curve becomes infinitely steep for a single instant, causing the slope limit to become ±∞\pm\infty. Since ∞\infty is not a real number, the derivative does not exist. Classic example: f(x)=x1/3f(x) = x^{1/3} at x=0x = 0.
  4. Discontinuity: If f(x)f(x) has a removable hole, jump, or vertical asymptote at x=ax = a, no unique tangent line can be constructed. Non-continuity automatically implies non-differentiability.

Engineering Applications

  • Civil & Transportation Engineering: Highway exit ramps, railway transitions, and roller coaster tracks must be designed with C1C^1-smooth continuity (continuous first derivative). A sharp corner (non-differentiable transition) would cause an instantaneous impulse spike in lateral centrifugal acceleration, leading to vehicle derailment or passenger injury.
  • Mechanical Stress Concentrations: Internal geometric corners in load-bearing structural brackets produce non-differentiable stress profiles, concentrating forces at the vertex. Mechanical engineers introduce smooth transition curves (fillets) to ensure differentiability and prevent fatigue fractures.

Example Problems

Example 1: Piecewise Boundary Differentiability Check

Determine if f(x)f(x) is differentiable at x=2x = 2:

f(x)={x2x≤24x−4x>2f(x) = \begin{cases} x^2 & x \le 2 \\ 4x - 4 & x > 2 \end{cases}

Solution
  1. Step 1: Check Continuity at x=2x = 2:
    • Left limit: lim⁡x→2−x2=4\lim_{x \to 2^-} x^2 = 4
    • Right limit: lim⁡x→2+(4x−4)=4(2)−4=4\lim_{x \to 2^+} (4x - 4) = 4(2) - 4 = 4
    • Function value: f(2)=4f(2) = 4 Since lim⁡x→2f(x)=f(2)=4\lim_{x \to 2} f(x) = f(2) = 4, f(x)f(x) is continuous at x=2x = 2.
  2. Step 2: Check One-Sided Derivatives at x=2x = 2:
    • Left-hand derivative: ddx[x2]∣x=2=2x∣x=2=4\frac{d}{dx}[x^2]\Big|_{x=2} = 2x\Big|_{x=2} = 4
    • Right-hand derivative: ddx[4x−4]∣x=2=4\frac{d}{dx}[4x - 4]\Big|_{x=2} = 4
  3. Conclusion: Since f(x)f(x) is continuous at x=2x = 2 and the left-hand derivative equals the right-hand derivative (4=44 = 4), f(x)f(x) is differentiable at x=2x = 2.

Connections & References