Find \(f_x, f_y, f_{xx}, f_{yy}, f_{xy}\) and \(f{yx}\) of:
\[ f(x,y)=y^3 +3xy^2 +3x^2y+x^3 \] #### First Partial Derivatives
To find the first partial derivatives \(f_x\) and \(f_y\), we can take the derivative of the function with respect to each of those variables, treating the other as a constant.
\[ f_x = 3y^2 + 6xy + 3x^2 \]
\[ f_y = 3y^2 + 6xy + 3x^2 \] Interesting! It appears to be by design that \(f_x\) and \(f_y\) look the same–that isn’t the case by default.
Same rules apply above–we’ll want to take the derivative of each derivative with respect to the designated variable, treating the other as a constant.
\[ f_{xx} = 6y + 6x \] \[ f_{yy} = 6y + 6x \]
Once again, this starting function is a special case resulting in \(f_{xx} = f_{yy}\)
These are more complicated, but only slightly. For \(f_{xy}\), we will take the derivative of \(f_x\), but this time with respect to \(y\). And we’ll do the reverse for \(f_{yx}\).
\[ f_{xy} = 6y + 6x \] \[ f_{yx} = 6y + 6x \]
Once again, these results mirroring each other are a fun idiosyncrasy of the original function, not an expectation for any two variable function!