Derivatives

Derivatives
Rules for finding derivatives
1. Constant Rule
The derivative of a constant is always zero. That is if then
2. Power Rule The derivative of a power function, Here is a number of any kind: integer, rational, positive, negative, even irrational, as in

Example 1:







3. Multiplication by constant: The derivative of is
4. Sum Rule: The derivative of is
5. Difference Rule: The derivative of is
6. Product Rule: The derivative of the product of two functions is not the product of the functions derivatives; rather, it is described by the equation below:


Example 2: Find derivative of the function

7. Quotient Rule: The derivative of the quotient of two functions is not the quotient of the functions derivatives; rather, it is described by the equation below:

Example 3: Find derivative of the function



8. Chain Rule:
Suppose that we have two functions and and they are both differentiable.
1.If we define then the derivative of is,

2.If we have and then the derivative of y is,

Example 4: Find derivatives of the functions













Higher Derivatives
If the derivative of a function exists in the domain of , then we have a new function. Now that we have agreed that the derivative of a function is a function, we can repeat the process and try to differentiate the derivative. The result, if it exists, is called the second derivative. It is denoted . The derivative of the second derivative is called the third derivative, written , and so on.
The nth derivative of is denoted . Thus
Leibniz’ notation for the nth derivative of is
Be careful to distinguish the second derivative from the square of the first derivative. Usually

Example 7: Find




Example 8: Compute the first, second and third derivatives of





Find derivative in each of the following problems



Compute the first, second and third derivatives in the following problems