Real Analysis – Differential Calculus I

— 7. Differential Calculus —

Definition 36 Let , and . is differentiable in point if the following limit exists This limit is represented by and is said to be the derivative of in .

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

On the other hand if we represent the time evolution of the position of a particle by the function the definition of its average velocity, on the time interval , is

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 is a smooth function then the limit exists and the it is the velocity of the particle:

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 . if , then one can define the right derivative in by

Let . if , then one can define the left derivative in by

If , exists iff and exist and are equal.

Definition 37 A function is said to be differentiable in if exists and is finite.

Definition 38 Let differentiable in . The map is called the derivative of and is represented by .

With the change of variable in definition 36 one can also define the derivative by the following expression:

Finally when can also denote the derivative of using Leibniz’s notation.

• is the increment along the axis
• is the increment along the axis

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

• is the infinitely small increment along the axis
• is the infinitely small increment along the axis

we can write the derivative as

As an example let us look into the function .

for all .

As another example we’ll now look into

for all .

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

• 
• 

Theorem 57 Let , and . If is differentiable in , there exists a continuous function and vanishing in such as:   Proof: Defining in the following way Since , then is null and vanishing in . To complete the proof one ha to check that with the previous construction of the identity of the theorem holds.

Corollary 58 Let differentiable in . Then it is when Proof: Let . Using Theorem 57 it is Since it is when .

Corollary 59 Let be differentiable in . Then is continuous in . Proof: From Theorem 57 it is

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:

with , a positive odd integer and .