Real Analysis – Differential Calculus I

— 7. Differential Calculus —

 Definition 36 Let ${D\subset\mathbb{R}}$, ${f:D\rightarrow\mathbb{R}}$ and ${c\in D\cap D'}$. ${f}$ is differentiable in point ${c}$ if the following limit exists $\displaystyle \lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c} \ \ \ \ \ (22)$ This limit is represented by ${f'(x)}$ and is said to be the derivative of ${f}$ in ${c}$.

The geometric interpretation of the value of the derivative is that it is the slope of the tangent of the curve that passes through ${c}$.

On the other hand if we represent the time evolution of the position of a particle by the function ${x=f(t)}$ the definition of its average velocity, on the time interval ${[t_0,t]}$, is

$\displaystyle v_a(t_0,t)=\frac{f(t)-f(t_0)}{t-t_0}$

If one is interested in knowing the velocity of a particle in a given instant, instead of knowing its average velocity in a given time interval, one has to take the previous definition and make the time interval as small as possible. If ${f}$ is a smooth function then the limit exists and the it is the velocity of the particle:

$\displaystyle v(t_0)=\lim_{t\rightarrow t_0}v_a(t_0,t)=\lim_{t\rightarrow t_0}\frac{f(t)-f(t_0)}{t-t_0}=f'(t_0)$

Hence the concept of derivative unifies two apparently different concepts:

1. The concept of the tangent to a curve. Which is a geometric concept.
2. The concept of the instantaneous velocity of a particle. Which is a kinematic concept.

The fact the two concepts that apparently have nothing in common are unified by a unique mathematical concepts is an indication of the importance of derivative.

Let ${f:D\rightarrow\mathbb{R}}$. if ${c\in D\cap D_{c^+}'}$, then one can define the right derivative in ${c}$ by

$\displaystyle f_+'(c)=\lim_{x\rightarrow c^+}\frac{f(x)-f(c)}{x-c}$

Let ${f:D\rightarrow\mathbb{R}}$. if ${c\in D\cap D_{c^-}'}$, then one can define the left derivative in ${c}$ by

$\displaystyle f_-'(c)=\lim_{x\rightarrow c^-}\frac{f(x)-f(c)}{x-c}$

If ${c\in D_{c^+}\cap D_{c^-}}$, ${f'(c)}$ exists iff ${f_+'(c)}$ and ${f_-'(c)}$ exist and are equal.

 Definition 37 A function ${f}$ is said to be differentiable in ${c}$ if ${f'(c)}$ exists and is finite.
 Definition 38 Let ${f:D\rightarrow\mathbb{R}}$ differentiable in ${D}$. The map ${x \in D \rightarrow f'(x)\in\mathbb{R}}$ is called the derivative of ${f}$ and is represented by ${f'}$.

With the change of variable ${h=x-c}$ in definition 36 one can also define the derivative by the following expression:

$\displaystyle f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Finally when can also denote the derivative of ${f}$ using Leibniz’s notation.

• ${\Delta x}$ is the increment along the ${x}$ axis
• ${\Delta f = f(x+h)-f(x)}$ is the increment along the ${y}$ axis

If one makes the increments infinitely small, that is to say if the increments are infinitesimals, then we denote them by:

• ${dx}$ is the infinitely small increment along the ${x}$ axis
• ${df}$ is the infinitely small increment along the ${y}$ axis

we can write the derivative as

$\displaystyle f'(x)=\frac{df}{dx}$

As an example let us look into the function ${f(x)=e^x}$.

{\begin{aligned} f'(x)&=\lim_{h\rightarrow 0}\dfrac{e^{x+h}-e^x}{h}\\ &=e^x\lim_{h\rightarrow 0}\dfrac{e^h-1}{h}\\ &=e^x \end{aligned}}

for all ${x\in\mathbb{R}}$.

As another example we’ll now look into ${f(x)=\log x}$

{\begin{aligned} f'(x)&=\lim_{h\rightarrow 0}\dfrac{\log (x+h)-\log x}{h}\\ &=\lim_{h\rightarrow 0}\dfrac{\log \left(x(1+h/x)\right)-\log x}{h}\\ &=\lim_{h\rightarrow 0}\dfrac{\log (1+h/x)}{h}\\ &=\lim_{h\rightarrow 0}\dfrac{h/x}{h}\\ &=1/x \end{aligned}}

for all ${x\in\mathbb{R}}$.

The following relationships are left as an exercise for the reader.

• ${(\sin x)'=\cos x}$
• ${(\cos x)'=-\sin x}$
 Theorem 57 Let ${D\subset\mathbb{R}}$, ${f:D\rightarrow\mathbb{R}}$ and ${c\in D\cap D'}$. If ${f}$ is differentiable in ${c}$, there exists a continuous function ${\varphi:D\rightarrow\mathbb{R}}$ and vanishing in ${c}$ such as: $\displaystyle f(x)=f(c)+\left( \left( f'(c)+\varphi(x) \right) (x-c) \right)\quad x\in D \ \ \ \ \ (23)$   Proof: Defining ${\varphi (x)}$ in the following way $\displaystyle f(x) = \begin{cases} \dfrac{f(x)-f(c)}{x-c}-f'(c) \quad \mathrm{if}\quad x \in D\setminus \{c\}\\ 0 \quad \mathrm{if}\quad x =c \end{cases}$ Since ${\displaystyle \lim_{x\rightarrow c}\varphi (x)=\lim_{x\rightarrow c} \left(\dfrac{f(x)-f(c)}{x-c}-f'(c)\right)=(f'(c)-f'(c)=0 }$, then ${\varphi}$ is null and vanishing in ${c}$. To complete the proof one ha to check that with the previous construction of ${\varphi}$ the identity of the theorem holds. $\Box$
 Corollary 58 Let ${f=D\rightarrow\mathbb{R}}$ differentiable in ${c}$. Then it is ${f(x)=f(c)+f'(c)(x-c)+o(x-c)}$ when ${x\rightarrow c}$Proof: Let ${r(x)=\varphi (x)(x-c)}$. Using Theorem 57 it is $\displaystyle f(x)=f(c)+f'(c)(x-c)+r(x)$ Since ${\lim_{x-to c}\varphi (x)=\varphi (c)=0}$ it is ${r(x)=o(x-c)}$ when ${x\rightarrow c}$. $\Box$
 Corollary 59 Let ${f}$ be differentiable in ${c}$. Then ${f}$ is continuous in ${c}$.Proof: From Theorem 57 it is {\begin{aligned} \lim_{x\rightarrow c} f(x)&=\lim_{x\rightarrow c}(f(c)+(f'(c)+\varphi (x))(x-c))\\ &=f(c) \end{aligned}} $\Box$

From corollary 59 it follows that all differentiable functions are continuous too. But is the converse also true? Is it true that all continuous functions are also differentiable?

The answer to the previous question is no. A simple example is the absolute value function:

An even more extreme example is the Weierstrass function:

$\displaystyle \sum_{n=0}^\infty a^n\cos\left( b^n\pi x \right)$

with ${0, ${b}$ a positive odd integer and ${ab>1+3/2\pi}$.