Maths-
General
Easy

Question

If u equals l o g invisible function application open parentheses x to the power of 3 end exponent plus y to the power of 3 end exponent plus z to the power of 3 end exponent minus 3 x y z close parentheses then sum x fraction numerator partial differential u over denominator partial differential x end fraction equals

  1. 1    
  2. 2    
  3. 3    
  4. 4    

hintHint:

We are given a function u. It is a function of three variables. We have to find the value of sum x fraction numerator partial differential u over denominator partial differential x end fraction. We will take the partial derivative of the function w.r.t to the variables and multiply with the respective variables. Then we will add them up.

The correct answer is: 3


    The given function is u=log(x3+y3+z3-3xyz)
    We have to find the value of sum x fraction numerator partial differential u over denominator partial differential x end fraction
    It means we have to find the value of x fraction numerator partial differential u over denominator partial differential x end fraction plus y fraction numerator partial differential u over denominator partial differential y end fraction plus z fraction numerator partial differential u over denominator partial differential z end fraction
    For simplification we will write u=logv
    v = x3 + y3 + z3 -3xyz
    We will take the partial derivative of u w.r.t all three variables.
    fraction numerator partial differential u over denominator partial differential x end fraction equals 1 over v fraction numerator partial differential v over denominator partial differential x end fraction
v space equals x cubed plus y cubed plus z cubed minus 3 x y z
y
fraction numerator partial differential v over denominator partial differential x end fraction space equals 3 x squared minus 3 y z space space space space space space space left parenthesis 1 right parenthesis

fraction numerator partial differential u over denominator partial differential y end fraction equals 1 over v fraction numerator partial differential v over denominator partial differential y end fraction
fraction numerator partial differential v over denominator partial differential y end fraction equals 3 y squared minus 3 x z space space space space space space space space left parenthesis 2 right parenthesis

fraction numerator partial differential u over denominator partial differential z end fraction equals 1 over v fraction numerator partial differential v over denominator partial differential z space end fraction
fraction numerator partial differential v over denominator partial differential z end fraction space equals 3 z squared minus 3 x y space space space space space space space space left parenthesis 3 right parenthesis






space
    Now, we will substitute all the values in the sum x fraction numerator partial differential u over denominator partial differential x end fraction.
    x fraction numerator partial differential u over denominator partial differential x end fraction plus y fraction numerator partial differential u over denominator partial differential y end fraction plus z fraction numerator partial differential y over denominator partial differential z end fraction equals x left parenthesis fraction numerator 3 x squared minus 3 y z over denominator x cubed plus y cubed plus z cubed minus 3 x y z end fraction right parenthesis plus y left parenthesis fraction numerator 3 y squared minus 3 x z over denominator x cubed plus y cubed plus z cubed minus 3 x y z end fraction right parenthesis plus z left parenthesis fraction numerator 3 z squared minus 3 x y over denominator x cubed plus y cubed plus z cubed minus 3 x y z end fraction right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 3 x cubed minus 3 x y z plus 3 y cubed minus 3 x y z plus 3 z cubed minus 3 x y z over denominator x cubed plus y cubed plus z cubed minus 3 x y z end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 3 x cubed plus 3 y cubed plus 3 z cubed minus 9 x y z over denominator x cubed plus y cubed plus z cubed minus 3 x y z end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 3 fraction numerator x cubed plus y cubed plus z cubed minus 3 x y z over denominator x cubed plus y cubed plus y cubed minus 3 x y z end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 3
    The required answer is 3.

    The variable v is used just for simplification. When we take partial derivative w.r.t to certain variable, we keep other variables constant.

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