The Derivative

The derivative measures the instantaneous rate of change of a function. Defined as the limit of difference quotients, it encodes information about tangent lines, velocity, and local behavior. Differentiability is stronger than continuity: every differentiable function is continuous, but not conversely. The derivative is the foundation of calculus and differential equations.

Definition

Definition5.1Derivative

Let $f : (a, b) \to \mathbb{R}$ and $c \in (a, b)$ . The derivative of $f$ at $c$ is

$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h},$

provided the limit exists. If $f'(c)$ exists, we say $f$ is differentiable at $c$ .

RemarkGeometric interpretation

$f'(c)$ is the slope of the tangent line to the graph of $f$ at $(c, f(c))$ . The equation of the tangent line is $y = f(c) + f'(c)(x - c)$ .

ExampleDerivative of x^n

For $f(x) = x^n$ (where $n \in \mathbb{N}$ ), we have $f'(x) = nx^{n-1}$ . Proof: using the binomial theorem,

$\frac{(x+h)^n - x^n}{h} = \frac{x^n + nx^{n-1}h + O(h^2) - x^n}{h} = nx^{n-1} + O(h) \to nx^{n-1}.$

ExampleDerivative of sin(x)

$\frac{d}{dx} \sin(x) = \cos(x)$ . Proof: using $\sin(x+h) = \sin(x)\cos(h) + \cos(x)\sin(h)$ and $\lim_{h \to 0} \sin(h)/h = 1$ , $\lim_{h \to 0} (\cos(h) - 1)/h = 0$ :

$\frac{\sin(x+h) - \sin(x)}{h} = \sin(x) \frac{\cos(h) - 1}{h} + \cos(x) \frac{\sin(h)}{h} \to 0 + \cos(x) = \cos(x).$

Differentiability implies continuity

Theorem5.1Differentiable implies continuous

If $f$ is differentiable at $c$ , then $f$ is continuous at $c$ .

Proof

We have

$f(c+h) - f(c) = \frac{f(c+h) - f(c)}{h} \cdot h \to f'(c) \cdot 0 = 0 \quad \text{as } h \to 0.$

Thus $\lim_{h \to 0} f(c+h) = f(c)$ , so $f$ is continuous at $c$ .

■

RemarkConverse is false

Continuity does not imply differentiability. For example, $f(x) = |x|$ is continuous at $x = 0$ but not differentiable there (the left and right derivatives are $-1$ and $+1$ , respectively).

Rules of differentiation

Theorem5.2Algebra of derivatives

If $f$ and $g$ are differentiable at $c$ , then:

$(f + g)'(c) = f'(c) + g'(c)$ (sum rule).
$(cf)'(c) = c f'(c)$ for any constant $c$ (scalar multiple rule).
$(fg)'(c) = f'(c)g(c) + f(c)g'(c)$ (product rule).
If $g(c) \neq 0$ , then $(f/g)'(c) = \frac{f'(c)g(c) - f(c)g'(c)}{g(c)^2}$ (quotient rule).

Theorem5.3Chain rule

If $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$ , then $f \circ g$ is differentiable at $c$ , and

$(f \circ g)'(c) = f'(g(c)) \cdot g'(c).$

ExampleChain rule application

Let $h(x) = \sin(x^2)$ . Then $h'(x) = \cos(x^2) \cdot 2x = 2x \cos(x^2)$ by the chain rule.

One-sided derivatives

Definition5.2One-sided derivatives

The right derivative of $f$ at $c$ is

$f'_+(c) = \lim_{h \to 0^+} \frac{f(c+h) - f(c)}{h}.$

Similarly, the left derivative is $f'_-(c) = \lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h}$ .

$f$ is differentiable at $c$ if and only if $f'_+(c) = f'_-(c)$ .

Examplef(x) = |x| at x = 0

For $f(x) = |x|$ , we have

$f'_+(0) = \lim_{h \to 0^+} \frac{|h|}{h} = 1, \quad f'_-(0) = \lim_{h \to 0^-} \frac{|h|}{h} = -1.$

Since $f'_+(0) \neq f'_-(0)$ , $f$ is not differentiable at $0$ .

Summary

The derivative is the fundamental notion of calculus:

Defined as the limit of difference quotients.
Differentiability implies continuity (but not conversely).
Rules: sum, product, quotient, chain rule.
Applications: tangent lines, rates of change, optimization.

See Mean Value Theorem and Taylor's Theorem for major applications.