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Topic 8: Differentiation
 

 

1. DEFINITIONS

The derivative of a function at point is given by the number

This number must be finite. In addition that limit must exist.

The left hand derivative of is the number and the right hand derivative is the number .

  • The function is differentiable at point if and only if it has both right hand and left hand derivatives at and they are equal.
  • A function is differentiable on an interval if it is differentiable at every point of that interval.

Example

FInd derivative of function   at

Solution

We know that

Since then is not differentiable at 0.

Notation

We have used the notation to denote the derivative of the function . There are also many other ways to denote the derivative like .

Theorem

If a function is differentiable on interval I then it is also continuous on that interval.

The converse of this theorem is not true.

Example

The function is continuous at but not differentiable at

2. DIFFERENTIALS

Consider a differentiable function . The differential of this function is denoted and defined by . From this relation we can write .

Example

If   then 

3. RULES FOR DIFFERENTIATIONS

3.1. Derivative of a sum

If are two differentiable functions, then

In fact,

3.2. Multiplication by a scalar

Consider a real number k and a differentiable function then .

In fact,

3.3. Derivative of a product

If are two differentiable functions, then

In fact,

3.4. Derivative of the inverse of a function

Consider a differentiable function different from zero then

In fact,

3.5. Derivative of a quotient

Given two differentiable functions then,

In fact,

3.6. Derivative of a power

Given a differentiable function then

In fact,

  • Verify for n = 1

          

           Then, it is true for n=2.

  • Suppose that the property is true for all n, it means
  • Verify that it is true for

         

          The property is true for , then it is true for all n

Thus,

Particular case

         

3.7. Derivative of a composite function

Given two differentiable functions then,

In fact,

3.8. Derivative of a reciprocal function

Recall that

4. DERIVATIVE OF TRIGONOMETRIC FUNCTIONS

4.1. Function sine and cosine

Function sine and cosine are differentiable on set of real numbers. In addition,

In fact,

Generally, if u is a differentiable function then

Also, recall that

Generally, if u is a differentiable function then

4.2. Function tangent and cotangent

The function tangent is differentiable on . In addition,

In fact,

Generally, if u is a differentiable function then

Also, recall that then

Generally, if u is a differentiable function then

4.3. Function arcsine and arccosine

 

4.4. Function arctangent and arccotangent

We know that

In the same way,

Generally, if u is a differentiable function then

5. SUCCESSIVE DERIVATIVES

Successive derivatives of a function are higher order derivatives of the same function.

The second derivative of function y = f  is

The third derivative of function y = f  is

The fourth derivative of function y = f  is

The nth derivative of function y = f  is

6. APPLICATIONS

6.1. Critical points

is a critical point of the functions if exists and if or

Example

Find all critical points of the function

Solution

The critical points are

6.2. Maximumu and minimum values

To find the maximum and the minimum values in given interval, follow the following steps:

  • find all critical point in the given interval
  • evaluate the function at the critical points found and at the end points of the given interval
  • identify the absolute the maximum and minimum values.

Example

Find the maximum and minimum values of the following functions

Solution

Now,

From this list

  • Maximum value is
  • Minimum values are

6.3. Increasing and decreasing of a function

 

6.4. Concavity of a function

 

6.5. L'Hospital rule

 

 

 
 

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