Calculate the second order linear derivatives fxy, fyx for the given function f(x,y)=x^3+8xy

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

I assume fyx means differentiation of f with respect to y followed by differentiation with respect to x, and vice versa for fyx.

fxy = d[d(x^3+8xy)/dx]/dy

=> fxy = d(3x^2 + 8y)/dy

=> fxy = 0 + 8

=> fxy = 8

fyx = d[d(x^3+8xy)/dy]/dx

=> fyx = d[0 + 8x]/dx

=> fyx = d[8x]/dx

=> fyx = 8

The required derivatives are fxy = 8 and fyx = 8.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll begin with fxy.

fxy = d^2f/dydx = [d(df/dx)]/dy

We'll differentiate with respect to x:

fxy = d[d(x^3+8xy)/dx]/dy

fxy = d(3x^2 + 8y)/dy

We'll differentiate with respect to x:

fxy = 8

We'll calculate fyx:

fyx = d^2f/dxdy = [d(df/dy)]/dx

fyx = d[d(x^3+8xy)/dy]/dx

fyx = d[d(8x)]/dx

fyx = 8

So, the second order linear derivatives are: fxy = 8 ; fyx = 8.

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