Algebra of derivative of function

Since derivatives are limits of functions, the rules for derivatives are almost the same as that of limits. So the algebra of derivatives is stated below without proof.

where is a constant.
.
This rule is known as product rule or Leibnitz Rule.

Product rule could also be remembered as

This is known as quotient rule.

Quotient rule could be remembered as
:

Example: 1
Differentiate from the first principles .

Solution:

Applying limit as we get
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Example: 2
Differentiate from the first principles

Solution:

Example: 3

Solution:
Here is a product of two functions Hence apply product rule.

Example: 4
Differentiate  with respect to .

Solution:

Example: 5
Differentiate

Solution:

Example: 6
Differentiate

Solution:

Example: 7
Find the derivative of

Solution:

Example: 8
If

Solution:

Example: 9

Solution: