Partial Differentiation - GATE Study 
                                   Material in PDF 
            We have learnt Differentiation in last topic. Now let's take a look at another concept in 
            Calculus – Partial Differentiation. These GATE 2019 Notes are important for 
            GATE EC, GATE EE, GATE ME, GATE CS, GATE CE as well as for other exams like IES, 
            BARC, BSNL, DRDO etc. This Study Material on Partial Differentiation can be 
            downloaded in PDF so that your preparation is made easy. Before you start, get basics in 
            Engineering Mathematics right. Moreover, you can solve online mock tests for exam 
            preparation. 
                        Event Details              Specifications 
                        Exam Name      GATE (Graduate Aptitude Test in Engineering) 
                        Conducting Body  IIT Madras 
                        Exam Level     National Level Examination 
                        Exam Mode      Online 
                        Exam Duration  180 minutes (3 hours) 
                        Language       English 
             
            Let f(x,y) be a function of two variables x and y.   
            The partial derivative of f(x,y) w.r.t x keeping ‘y’ as constant is defined as   
                                     
            Similarly, the partial derivative of δ=f(x,y)  w.r. to y keeping ‘x’ as constant is defined as   
                                      
            1. Partial differentiation is nothing but ordinary differentiation only treating one of the 
            variables as constant.    
                                                                         .   
            3. Geometrically partial differentiation represents equation of surface where as 
            ordinary differentiation represents equation of curve.   
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                         Limit of a Function of Two Variables   
                         A function f(x, y) is said to be tend to the limit l as (x, y) tends to (a, b) (i.e.) x → a and 
                         y → b if corresponding to any given positive number ε there exists a positive number δ 
                         such |f(x,y) - l| < ε for all points (x, y)  whenever |x-a| ≤ δ,   |y-b| ≤ δ   
                         In other words the variable value f(x, y) approaches finite fixed value l in the codomain 
                         when the variable value (x, y) approaches a fixed value (a, b) i.e. x approaches a and y 
                         approaches b simultaneously. We write it as   
                                                                                                       
                           
                         Example 1:   
                                                                         
                         Solution:    
                                                                 
                                                                 
                         So limit exists.   
                         Continuity of a Function of Two Variables at a Point   
                         f(x,y) is said to be continuous at (a, b) on its domain of definition if   
                                                                      
                           
                         Example 2:   
                                                                                                   
                         Solution:   
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                                     x −y                −y                                            
                             lim            = lim             = − 1   _ _ _ _ _ _ _ _ _ _ _ ( 2 ) 
                             y → 0 x +y          y → 0  y                                                  
                             x → 0 
                             (1)  ≠  (2) 
                                                  
                                  Limit does not exist.                         
                               
                             Example 3:   
                                                                                                                                                                                      
                             Solution:   
                                                                                            
                                                                                            
                                                                                                                  
                                                                                      
                                                                             
                               
                             Second Order Partial Differentiation   
                              Let z = f(x,y) be a given function and Differentiated partially w.r.t x and y separately   
                                                                                                                  .   
                               
                             Standard Notations:   
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                     Example 4: 
                               m             2      2      2 
                     If U = r      where r    =  x    +  y    then find the value of U            +  U      
                                                                                               xx      yy                                          
                    Solution:   
                                                                  
                                                       
                                                                               
                                                                                              
                                                                                           
                                                                        
                       
                    Homogeneous Function    
                                                                         a0                                       a1      a2                    an-1
                                                                       an                                              
                                    
                                                                                                                                                           
                        
                     Euler’s Theorem:     
                                                                                                                                      
                       
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