Derivative at a Point

Basis
Last updated: Tags: Calculus, Differentiation

Prerequisites

The tangent line at a point captures the instantaneous slope of a function. This checkpoint makes that intuition precise: the derivative of ff at a point x0x_0 is the limit of the difference quotient, and the formal definition gives it a firm foundation.

Definition

Definition. Let ff be defined on an open interval containing x0x_0. The derivative of ff at x0x_0, written f′(x0)f'(x_0), is

f′(x0)  =  lim⁡h→0f(x0+h)−f(x0)h,(1)f'(x_0) \;=\; \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}, \tag{1}

provided this limit exists and is finite. When it does, ff is said to be differentiable at x0x_0.

Equivalently, via the substitution x=x0+hx = x_0 + h:

f′(x0)  =  lim⁡x→x0f(x)−f(x0)x−x0.f'(x_0) \;=\; \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}.

The notations dfdx∣x=x0\dfrac{df}{dx}\big|_{x=x_0} and ddxf(x)∣x=x0\dfrac{d}{dx}f(x)\big|_{x=x_0} are also standard.

Computing from first principles

Power function f(x)=xnf(x) = x^n, n∈Nn \in \mathbb{N}

By the binomial theorem,

(x0+h)n−x0nh  =  nx0n−1+(n2)x0n−2h+⋯+hn−1.\frac{(x_0+h)^n - x_0^n}{h} \;=\; n x_0^{n-1} + \binom{n}{2} x_0^{n-2} h + \cdots + h^{n-1}.

Every term except the first contains a factor of hh, so as h→0h \to 0 the quotient converges to nx0n−1nx_0^{n-1}:

ddx(xn)  =  nxn−1.\frac{d}{dx}(x^n) \;=\; n x^{n-1}.

Constant function f(x)=cf(x) = c

The difference quotient is c−ch=0\dfrac{c - c}{h} = 0 for all h≠0h \neq 0, so f′(x0)=0f'(x_0) = 0.

Differentiability implies continuity

Theorem. If ff is differentiable at x0x_0, then ff is continuous at x0x_0.

Proof. Write f(x0+h)−f(x0)=f(x0+h)−f(x0)h⋅hf(x_0 + h) - f(x_0) = \dfrac{f(x_0+h)-f(x_0)}{h} \cdot h. The first factor converges to f′(x0)f'(x_0) and the second to 00. By the product rule for limits,

lim⁡h→0[f(x0+h)−f(x0)]  =  f′(x0)⋅0  =  0,\lim_{h \to 0}\bigl[f(x_0+h) - f(x_0)\bigr] \;=\; f'(x_0) \cdot 0 \;=\; 0,

so lim⁡h→0f(x0+h)=f(x0)\lim_{h\to 0} f(x_0+h) = f(x_0). □\square

The converse fails: f(x)=∣x∣f(x) = |x| is continuous at 00 but is not differentiable there — the left and right difference quotients approach −1-1 and +1+1 respectively.

One-sided derivatives

Definition. The right derivative and left derivative at x0x_0 are

f+′(x0)  =  lim⁡h→0+f(x0+h)−f(x0)h,f−′(x0)  =  lim⁡h→0−f(x0+h)−f(x0)h.f'_+(x_0) \;=\; \lim_{h \to 0^+}\frac{f(x_0+h)-f(x_0)}{h}, \qquad f'_-(x_0) \;=\; \lim_{h \to 0^-}\frac{f(x_0+h)-f(x_0)}{h}.

The two-sided derivative f′(x0)f'(x_0) exists if and only if both one-sided derivatives exist and are equal.

Summary

  • f′(x0)=lim⁡h→0f(x0+h)−f(x0)hf'(x_0) = \lim_{h\to 0}\dfrac{f(x_0+h)-f(x_0)}{h}; if this limit exists, ff is differentiable at x0x_0.
  • (xn)′=nxn−1(x^n)' = nx^{n-1}, derived from the binomial theorem.
  • Differentiability implies continuity, but not vice versa.
  • f′(x0)f'(x_0) exists if and only if f+′(x0)f'_+(x_0) and f−′(x0)f'_-(x_0) both exist and are equal.