# Real Analysis/Differentiation

←Exercises | Real AnalysisDifferentiation |
Applications of Derivatives→ |

## Definition[edit]

We are now ready to define the derivative of a function

Let

Let

We say that *ƒ*(*x*) is **differentiable** at *x*=*a* if and only if there exists a real number *L* such that

.

*L* is said to be the **derivative** of *ƒ* at *a* and is denoted by

A function is said to be differentiable on a set *A* if the derivative exists for each *a* in *A*. A function is differentiable if it is differentiable on its entire domain.

Conceptually, finding the derivative means finding the slope of the tangent line to the function. Thus the derivative can be thought of as a linear, or first-order, approximation of the function.

## Properties[edit]

Some properties of the derivative follow immediately from the definition:

### Basic Properties[edit]

If f and g are differentiable, then:

#### Proof[edit]

### Theorem(Differentiability Implies Continuity)[edit]

If f is differentiable at x, it is continuous at x

#### Proof[edit]

Since f is differentiable at x, .

So

Thus , so f is continuous at x.

### Theorem(Product Rule)[edit]

If f and g are differentiable, then

#### Proof[edit]

, since g is continuous at x.

The following theorem is a bit trickier to prove than it seems. We would like to use the following argument:

The problem is that may be zero at points arbitrarily close to x, and therefore would not be continuous at these points. Thus we apply a clever lemma as follows:

### Theorem (Caratheodory's Lemma)[edit]

Let

We say that is differentiable at if and only if there exists a continuous function that satisfies

#### Proof[edit]

()Let be differentiable at and define function such that

for and

It is easy to see that is continuous and that it satisfies the required condition.

() Let be a continuous function satisfying

For all , we have that

As is continuous, , that is,

which implies that is differentiable at

### Theorem(Chain Rule)[edit]

Let be differentiable at

Let be differentiable at

Let the domain of f be a subset of the image of g.

Then

(i) is differentiable at

(ii)

#### Proof[edit]

Caratheodory's Lemma implies that there exist continuous functions such that and

Now, consider the function . Obviously, is continuous.

Also, it satisfies . Hence, by Caratheodory's Lemma, is differentiable at and that

## Examples[edit]

Consider by . What is the derivative of at ?

So, here we see that . Since was an arbtarily choosen point we conculde that

Similar derivative formula's may also be found.

## Exercises[edit]

- Find the derivatives of the common functions: Polynomial, Trigonometric, Exponential and Logarithmic.
- Some of the most popular counter examples to illustrate properties of continuity and differentiability are functions involving

(i)Prove that is not continuous at

(ii)Prove that the function is continuous but not differentiable at

(iii)Prove that is differentiable at