2.8 Notes on the Derivative as a Function and Higher Derivatives

The Derivative Function

  • Definition: The derivative at x is
    f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}
    for those x where the limit exists.
  • Interpretation: f'(x) is the slope of the tangent to the graph of f at (x, f(x)). The derivative function f' collects these slopes for all x where the limit exists.
  • Domain note: The domain of f' may be smaller than the domain of f, since the limit may not exist at some x.

Notation and Interpretation

  • Common notations: f'(x), \dfrac{dy}{dx}, \dfrac{df}{dx}, Df(x), D f(x), \dfrac{d}{dx}f(x).
  • The symbols D and d/dx are differentiation operators.
  • Leibniz notation: \dfrac{dy}{dx} is a useful symbol, but it is not to be interpreted strictly as a ratio; it is a way to denote the derivative. It is also convenient when used with increments: if y=f(x), then dy and dx represent small changes.

Differentiability and Continuity

  • A function f is differentiable at a if f'(a) exists.
  • Differentiable on an open interval means differentiable at every point in that interval.
  • Key theorem: If f is differentiable at a, then f is continuous at a. (The converse is false.)

Example: Absolute Value

  • Consider f(x)=|x|.
  • Differentiability:
    • For x>0: f'(x)=1
    • For x<0: f'(x)=-1
    • At x=0: f' does not exist (corner in the graph).

How a Function Can Fail to Be Differentiable

  • If the graph has a "corner" or "kink" at a, left and right derivatives disagree.
  • If f is not continuous at a, then f is not differentiable at a (discontinuities).
  • If the graph has a vertical tangent at a (continuous at a but tangent tends to vertical), then f'(a) does not exist (dx tends to 0 too fast).

Higher Derivatives

  • If f is differentiable, then f' is a function and may itself be differentiable. The second derivative is
    f''(x)=\frac{d}{dx}\big(f'(x)\big)=\frac{d^2 f}{dx^2}.
  • Notation for higher derivatives: f^{(n)}(x)=\frac{d^n f}{dx^n}.
  • Interpretation:
    • f''(x) is the rate of change of the slope, e.g., acceleration in physics when f is a position function s(t).
    • In Leibniz form for a position function s(t):
      v(t)=\frac{ds}{dt},\quad a(t)=\frac{dv}{dt}=\frac{d^2 s}{dt^2}.
  • Third derivative and beyond:
    • f'''(x)=\frac{d}{dx}(f''(x))=\frac{d^3 f}{dx^3}.
    • In motion terms, the third derivative of position is the jerk:
      j(t)=\frac{d^3 s}{dt^3}.
  • General pattern: the nth derivative is f^{(n)} and can be denoted in various notations (e.g., y^{(n)} or d^n y/dx^n).

Notation and Quick Reference

  • Primary notations:
    • f'(x) or dy/dx for the first derivative.
    • f''(x) or d^2 y/dx^2 for the second derivative.
    • f^{(n)}(x) for higher derivatives, with d^n y/dx^n as Leibniz notation.
  • In physics and geometry, derivatives have direct interpretations (slope, velocity, acceleration, jerk).

Quick Takeaways

  • The derivative is a function (where defined) describing slopes of tangents.
  • Differentiability at a point implies continuity there.
  • Common failure modes: corners, discontinuities, vertical tangents.
  • Higher derivatives measure rates of change of rates of change; they include velocity, acceleration, jerk, etc.
  • Notation: f'(x), f''(x), f^{(n)}(x); Leibniz form: dy/dx, d^n y/dx^n; D as a differentiation operator.