Chapter 12.3, Pg 711 Question 10 find \(f_x, \ f_y, \ f_{xx}, \ f_{yy}, \ f_{xy}, \
f_{yx}\)
Where \(f(x,y) = x^2y + 3x^2 + 4y -
5\).
The partial derivatives are given as:
\[
\begin{split}
f_x &= \frac{\partial f}{\partial x} = 2xy + 6 \\
f_y &= \frac{\partial f}{\partial y} = x^2 + 4 \\
f_{xx} &= \frac{\partial^2 f}{\partial x^2} =\frac{\partial
}{\partial x} (\frac{\partial f}{\partial x}) = 2y + 6 \\
f_{yy} &= \frac{\partial^2 f}{\partial y^2} =\frac{\partial
}{\partial y} (\frac{\partial f}{\partial y}) = 0 \\
f_{xy} &= \frac{\partial^2 f}{\partial x \partial y} =\frac{\partial
}{\partial x} (\frac{\partial f}{\partial y}) = 2x \\
f_{yx} &= \frac{\partial^2 f}{\partial y \partial x} =\frac{\partial
}{\partial y} (\frac{\partial f}{\partial x}) = 2x \\
\end{split}
\] Note: \(f_{xy} = f_{yx} \ \
=>\ \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2
f}{\partial y \partial x}\)
i.e Mixed partial derivatives are equal.