Differentiation (One Variable)

The fourth page of the analysis spine. We define the derivative, establish its rules, and prove the Mean Value Theorem — the workhorse that connects the derivative to the global behaviour of a function — together with Taylor's theorem. These results are the one-dimensional seeds of the multivariable derivative (multivariable.md) and of the Fundamental Theorem of Calculus (riemann-integral.md).

References: Rudin, Principles of Mathematical Analysis, ch. 5; Spivak, Calculus.

1. The Derivative

Let $f$ be defined on an open interval containing $c$. The derivative of $f$ at $c$ is

$$f^{\prime }(c)=\underset{h\to 0}{\lim }\frac{f(c+h)-f(c)}{h}=\underset{x\to c}{\lim }\frac{f(x)-f(c)}{x-c},$$

when the limit exists; then $f$ is differentiable at $c$. The geometric content is the slope of the tangent line; the analytic content is local linear approximation:

$$f(c+h)=f(c)+f^{\prime }(c)h+o(h),\text{where }o(h)/h\to 0.$$

This "best linear approximation" viewpoint — not the difference quotient — is what generalizes to the total derivative in several variables (multivariable.md §2).

Differentiable $\Rightarrow$ continuous. If $f$ is differentiable at $c$ it is continuous there (the increment $f(c+h)-f(c)=f^{\prime }(c)h+o(h)\to 0$). The converse fails: $|x|$ is continuous but not differentiable at $0$, and there exist functions continuous everywhere and differentiable nowhere (Weierstrass).

2. Rules of Differentiation

For differentiable $f,g$:

• Linearity: $(\alpha f+\beta g{)}^{\prime }=\alpha f^{\prime }+\beta g^{\prime }$.
• Product (Leibniz): $(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$.
• Quotient: $(f/g{)}^{\prime }=(f^{\prime }g-f{g}^{\prime })/g^2$ where $g\ne 0$.
• Chain rule: if $g$ is differentiable at $c$ and $f$ at $g(c)$, then $(f\circ g{)}^{\prime }(c)=f^{\prime }(g(c))g^{\prime }(c)$.
• Inverse function (1-D): if $f$ is differentiable with $f^{\prime }(c)\ne 0$ and a continuous inverse near $f(c)$, then $(f^{-1}{)}^{\prime }(f(c))=1/f^{\prime }(c)$.

The chain rule is cleanest in the linear-approximation picture: the best linear approximation of a composite is the composite of the best linear approximations — exactly the statement that generalizes to Jacobian matrices multiplying.

3. Local Extrema and Rolle's Theorem

If $f$ has a local extremum at an interior point $c$ and is differentiable there, then $f^{\prime }(c)=0$ (Fermat's interior extremum theorem) — at a peak or trough the tangent is horizontal. Points with $f^{\prime }(c)=0$ are critical points.

Rolle's Theorem. If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then $f^{\prime }(\xi )=0$ for some $\xi \in (a,b)$.

Proof. By the EVT (continuity.md) $f$ attains a max and min on $[a,b]$. If both are at endpoints then $f$ is constant and $f^{\prime }\equiv 0$; otherwise an extremum is interior, and Fermat gives $f^{\prime }(\xi )=0$. $\square$

4. The Mean Value Theorem

Mean Value Theorem (MVT). If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there is $\xi \in (a,b)$ with $$f^{\prime }(\xi )=\frac{f(b)-f(a)}{b-a}.$$

Proof. Apply Rolle to $g(x)=f(x)-\frac{f(b)-f(a)}{b-a}(x-a)$, which satisfies $g(a)=g(b)=f(a)$. $\square$

The MVT converts local derivative information into global statements:

• $f^{\prime }>0$ on an interval $\Rightarrow f$ strictly increasing; $f^{\prime }=0\Rightarrow f$ constant; $f^{\prime }\ge 0\Rightarrow$ nondecreasing.
• Lipschitz bound: $|f^{\prime }|\le M\Rightarrow |f(x)-f(y)|\le M|x-y|$.
• It is the key lemma in proving the Fundamental Theorem of Calculus (riemann-integral.md §5).

Cauchy's generalized MVT. For $f,g$ as above, $(f(b)-f(a))g^{\prime }(\xi )=(g(b)-g(a))f^{\prime }(\xi )$ for some $\xi$. This yields L'Hôpital's rule for $0/0$ and $\infty /\infty$ indeterminate limits.

5. Higher Derivatives and Smoothness Classes

If $f^{\prime }$ is itself differentiable we write $f^{\prime \prime }$, and inductively $f^{(k)}$. Define:

• $f\in C^k(I)$ if $f^{(k)}$ exists and is continuous on $I$;
• $f\in C^{\infty }(I)$ (smooth) if derivatives of all orders exist;
• $f$ (real-)analytic if it equals its Taylor series locally (§6).

The inclusions are strict: $C^0⊋C^1⊋\cdots ⊋C^{\infty }⊋C^{\omega }$ (analytic). The standard counterexample $f(x)=e^{-1/x^2}$ (with $f(0)=0$) is smooth but not analytic at $0$: every derivative vanishes there, so its Taylor series is identically $0\ne f$. This $C^{\infty }$-vs-analytic gap is why bump functions and partitions of unity exist — the technical backbone of integration on manifolds in stokes-general.md.

6. Taylor's Theorem

Taylor's Theorem (Lagrange remainder). If $f\in C^n[a,b]$ and $f^{(n+1)}$ exists on $(a,b)$, then for $x,c\in [a,b]$, $$f(x)=\sum _{k=0}^n\frac{f^{(k)}(c)}{k!}(x-c{)}^k+R_n(x),>R_n(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}(x-c{)}^{n+1}$$ for some $\xi$ between $c$ and $x$.

The polynomial part is the $n$-th Taylor polynomial; $R_n$ is the remainder. The proof is a repeated application of the generalized MVT. Other remainder forms (integral, Cauchy) follow from the FTC. Taylor's theorem quantifies how well polynomials approximate smooth functions and is the one-variable model for the multivariable Taylor expansion (multivariable.md §5) used in second-derivative tests.

A function is analytic at $c$ when $R_n(x)\to 0$ on a neighbourhood, so that $f(x)={\sum }_{k\ge 0}\frac{f^{(k)}(c)}{k!}(x-c{)}^k$ — a convergent power series (sequences-series.md §7.1).

7. Differentiating Limits

Combining with sequences-series.md §7: if $f_n\to f$ pointwise and $f_n^{\prime }\to g$ uniformly on an interval, then $f$ is differentiable and $f^{\prime }=g$. Uniform convergence of the derivatives (not of the functions) is the correct hypothesis — this is what legitimizes term-by-term differentiation of power series within their radius of convergence.

8. Where this page is used

• MVT is the lemma behind the Fundamental Theorem of Calculus in riemann-integral.md.
• Linear-approximation view of $f^{\prime }$ generalizes to the total derivative and Jacobian in multivariable.md.
• Smoothness classes / non-analytic bump functions underpin partitions of unity used to integrate forms in stokes-general.md.

Next: The Riemann Integral.