Differentiation: Theory and Principles

Differentiation is the study of the local behavior of functions. In Analysis, we move beyond the mechanics of “power rules” to investigate the structural requirements for a function to be smooth and the profound theorems that relate the derivative to the function’s overall shape.

The Definition of the Derivative

Let $f:I\to \mathbb{R}$ be a function defined on an open interval $I$. The derivative of $f$ at $c\in I$ is the limit: $f^{\prime }(c)={\lim }_{h\to 0}\frac{f(c+h)-f(c)}{h}$ If this limit exists, we say $f$ is differentiable at $c$.

Continuity vs. Differentiability

A classic theorem in analysis states that if $f$ is differentiable at $c$, then $f$ is continuous at $c$. Proof Sketch: $f(c+h)-f(c)=\left[\frac{f(c+h)-f(c)}{h}\right]\cdot h$. As $h\to 0$, the first term approaches $f^{\prime }(c)$ and the second term approaches $0$, so the product approaches $0$, ensuring ${\lim }_{x\to c}f(x)=f(c)$. Crucially, the converse is false. The Weierstrass function is continuous everywhere but differentiable nowhere.

Local Linear Approximation

The existence of a derivative at $c$ means that $f(x)$ can be approximated by a linear function near $c$: $f(x)\approx f(c)+f^{\prime }(c)(x-c)$ The error in this approximation, $E(x)=f(x)-[f(c)+f^{\prime }(c)(x-c)]$, satisfies ${\lim }_{x\to c}\frac{E(x)}{x-c}=0$. This perspective is essential for generalizing derivatives to higher dimensions (linear maps).

Core Theorems of Differentiable Functions

1. Rolle’s Theorem

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists at least one $c\in (a,b)$ such that $f^{\prime }(c)=0$.

2. The Mean Value Theorem (MVT)

This is arguably the most important theorem in differential calculus. If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in (a,b)$ such that: $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$ Significance: The MVT links local information ($f^{\prime }$) to global information ($f(b)-f(a)$). It is used to prove that if $f^{\prime }>0$, the function is increasing.

3. Darboux’s Theorem

Despite what one might expect, derivatives do not have to be continuous. However, they must satisfy the Intermediate Value Property. If $f$ is differentiable, $f^{\prime }$ takes on every value between $f^{\prime }(a)$ and $f^{\prime }(b)$.

Differentiation Rules in Analysis

While the formulas are familiar, their validity depends on the differentiability of the components:

• Chain Rule: $(f\circ g{)}^{\prime }(c)=f^{\prime }(g(c))g^{\prime }(c)$.
• Inverse Function Theorem: If $f$ is $C^1$ and $f^{\prime }(c)\ne 0$, then $f$ is locally invertible, and $(f^{-1}{)}^{\prime }(f(c))=1/f^{\prime }(c)$.

L’Hôpital’s Rule and Indeterminate Forms

Analytical rigor is required to use L’Hôpital’s Rule. For functions $f,g$ such that $\lim f(x)=\lim g(x)=0$ (or $\infty$): ${\lim }_{x\to c}\frac{f(x)}{g(x)}={\lim }_{x\to c}\frac{f^{\prime }(x)}{g^{\prime }(x)}$ provided the latter limit exists. This rule allows for the evaluation of limits that are otherwise algebraically inaccessible.

Smoothness Classes $C^n$

Functions are classified by how many continuous derivatives they possess:

• $C^0$: Continuous.
• $C^1$: Differentiable, and the derivative is continuous.
• $C^n$: $n$ continuous derivatives.
• $C^{\infty }$: Smooth (infinitely differentiable).
• $C^{\omega }$: Analytic (can be represented by a power series).

Python: Symbolic Calculus

In university-level math, we often use symbolic computation to verify identities or find exact derivatives.

import sympy as sp

# Define variable and function
x = sp.Symbol('x')
f = sp.ln(sp.sin(x)**2 + 1)

# Compute First and Second Derivatives
f1 = sp.diff(f, x)
f2 = sp.diff(f1, x)

print(f"f'(x): {f1}")
print(f"f''(x): {sp.simplify(f2)}")

# Compute Taylor Expansion at x=0
f_series = sp.series(f, x, 0, 5)
print(f"Taylor Series: {f_series}")

Summary

Differentiation is not just about the slope of a tangent line; it is about the local approximation of non-linear phenomena. By understanding the conditions under which a derivative exists and the theorems (like MVT and Darboux) that govern its behavior, we gain the ability to analyze the dynamics of the world with mathematical certainty.

Exercise