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,
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
