The Derivative

Near a point where it is well-behaved, the graph of a function looks like a straight line, and that line has a slope recording the rate at which the function's value changes. Analysis makes this precise without appealing to the picture: the slope is a limit of a function, and every rule of differential calculus becomes a theorem about that limit rather than a fact read off a drawing.

The difference quotient and its limit

Fix an interval $I$, a function $f:I\to \mathbb{R}$, and a point $c\in I$. The difference quotient of $f$ based at $c$ is

It is the slope of the secant line through the two graph points $(c,f(c))$ and $(x,f(x))$. As $x$ moves toward $c$ the second point slides along the graph toward the first, and the secant pivots. If those secant slopes approach a single number, that number is the slope of the tangent line, and it is the derivative.

Two conventions are worth stating once. First, $c$ is allowed to be an endpoint of $I$; the limit is then automatically one-sided, and every theorem below still holds, which saves the trouble of excising endpoints. Second, the derivative is sometimes written $\frac{\mathbf{d}f}{\mathbf{d}x}$ or $\frac{\mathbf{d}}{\mathbf{d}x}f(x)$; the notations name the same object.

Worked derivatives from the definition

Every basic derivative is a limit computation: simplify the difference quotient algebraically until the $x-c$ in the denominator cancels, then let $x\to c$.

For an affine function $f(x)=ax+b$ the quotient is $\frac{a(x-c)}{x-c}=a$, so $f^{\prime }(c)=a$ at every $c$. Every differentiable function behaves infinitesimally like an affine function, so working a formula out for $ax+b$ first is a reliable way to guess the general rule.

The increment form and one-sided derivatives

Substituting $x=c+h$ recenters the difference quotient on the increment $h$:

The two forms are identical, but the increment form makes the two-sidedness explicit. The limit exists only if the right-hand derivative (letting $h\to 0^{+}$) and the left-hand derivative ($h\to 0^{-}$) both exist and agree. When $c$ is an endpoint of $I$ only one side is available, which is why endpoints need no special treatment. When both sides exist but differ, $f$ has a corner and is not differentiable, the situation of $|x|$ at the origin below.

Differentiability forces continuity

A function cannot have a jump or a hole where it is differentiable: the finite limit defining $f^{\prime }(c)$ cannot exist across a break.

The converse fails: continuity is a much weaker property than differentiability.

Continuous functions with no derivative

The absolute value is the first counterexample.

The absolute value fails at a single point. Weierstrass exhibited a function that is continuous on all of $\mathbb{R}$ yet differentiable at no point at all: a graph with a corner everywhere and a tangent nowhere. Its construction rests on uniformly convergent series. Continuity places no bound on how badly a function can fail to be smooth.

The algebra of derivatives

Differentiation is linear, and it interacts with products, quotients, and compositions through fixed rules. Each rule is proved by manipulating difference quotients and passing to the limit, so each rests on the limit laws already established for functions.

Linearity

The product rule

The derivative of a product is not the product of the derivatives. The correct statement is the product rule, and its shape is visible in a picture: an increment in $fg$ is an $f$-strip plus a $g$-strip on a rectangle of sides $f$ and $g$.

The proof rests on the algebraic identity

which decomposes the shaded region below. Dividing by $x-c$, taking $x\to c$, and using continuity of $f$ at $c$ (so $f(x)\to f(c)$) gives the stated formula.³

Strip $A$ has area $f(x)(g(x)-g(c))$ and strip $B$ has area $(f(x)-f(c))g(c)$; roughly, $\Delta (fg)=f\Delta g+\Delta fg$, and dividing by $\Delta x$ and shrinking it recovers the rule.

The quotient rule

The quickest derivation differentiates $1/x$ from the definition, then combines the reciprocal with the chain and product rules; the direct difference-quotient computation also works.

The chain rule

Composition is how complicated functions are built, and the chain rule differentiates a composition. Its content is that local linear approximations compose by multiplying their slopes: if $g$ scales small increments near $c$ by $g^{\prime }(c)$ and $f$ scales small increments near $g(c)$ by $f^{\prime }(g(c))$, then $f\circ g$ scales by the product.

The device of patching the difference quotient into a continuous function $u$ is what avoids the false step of dividing by $g(x)-g(c)$, which can be zero even when $x\ne c$.

Powers and polynomials

The rules together compute the derivative of every power of $x$ by induction, which in turn differentiates every polynomial.

A polynomial $p(x)={\sum }_{k=0}^d{a}_k{x}^k$ is a finite linear combination of powers, so linearity and the power rule give $p^{\prime }(x)={\sum }_{k=1}^{d}k{a}_k{x}^{k-1}$ at every $x$. Rational functions follow from the quotient rule on their domains. Between the four algebraic rules and the power rule, every function built by finite arithmetic from $x$ is differentiable and its derivative is mechanical.

Reading the rules as one table

The four rules turn any expression built from known derivatives into a mechanical computation. Each is an identity about the derivative at a point where the ingredients are differentiable.

Applied repeatedly, these give the derivative of every polynomial ($\frac{\mathbf{d}}{\mathbf{d}x}{x}^n=n{x}^{n-1}$ from the product rule and induction), every rational function on its domain, and every algebraic combination of functions whose derivatives are already known.