General Theorems of Differentiation:

Theorem:
            The derivative of a constant function is zero.
Proof:
             Let
I.                             
II.                           
                               
                               
III.                          
                               
IV.                          
                                               

                                               

Theorem (Power Formula):
            
Proof:
            Let
I.             
II.           
               
               
               
By using the expansion of binomial series
               
                              
                               
                            
III.                          
                                               
IV.                          
                                               
                                               

                               

Theorem:
            If 'c' is any constant, then.
Proof:
            Let
I.                             
II.                           
                               
                               
III.                          
IV.                          
                                               
                                               

                                               

Theorem:
            The derivative of sum of two functions is the sum of their derivatives.
Proof:
            Let
I.                                             
II.                                           
                                               
                                                 
                                               
III.                                          
                                                               
IV.                                          
                                                               

                                               

Theorem:
            The derivative of the difference of two functions is the difference of their derivatives.
Proof:
            Let
I.                             
II.                           
                              
                               
                               
III.                          
                                               
IV.                          
                                               

                                               

Theorem: (Product Rule)
            
Proof:
            Let
I.                             
II.                           
                                 
                                                Subtracting & Adding
                               
                               
III.          
               
IV.          
                               
                               

                               

Theorem (Quotient Rule):
            
Proof:
            Let                          
I.                             
III.                          
                               
                               
Subtracting & Adding
                               
                               
                               
III.          
                               
                      
IV.          
                               
               
               

               

Chain Rule:
            If andare two differentiable functions, then the derivative of the composite functionis given by
            
Proof:
            Let, andbe the increments of x, u, and y respectively. Then from algebra, we have
(however small ,andmay be)
Proceeding to limit whenand consequentlyand, therefore also approaches zero, we have

            Hence     

(Examples of Derivative)

Differential Calculus

(Examples of General Theorems)