Sure! Here's a brief explanation and solution for each problem:

4. $f(x,y) = x^2 - 2xy + 3y^2$

• Partial derivative with respect to $x$: $f_x = \frac{\partial}{\partial x}(x^2 - 2xy + 3y^2) = 2x - 2y$

• Partial derivative with respect to $y$: $f_y = \frac{\partial}{\partial y}(x^2 - 2xy + 3y^2) = -2x + 6y$

5. $f(x,y) = xe^{-y} + ye^{-2x}$

• Partial derivative with respect to $x$: $f_x = \frac{\partial}{\partial x}(xe^{-y} + ye^{-2x}) = e^{-y} - 2ye^{-2x}$

• Partial derivative with respect to $y$: $f_y = \frac{\partial}{\partial y}(xe^{-y} + ye^{-2x}) = -xe^{-y} + e^{-2x}$

6. $z = \frac{1}{\ln(x^2 + y^2)}$

• Partial derivative with respect to $x$: $z_x = \frac{\partial}{\partial x}\left(\frac{1}{\ln(x^2 + y^2)}\right) = -\frac{2x}{(x^2 + y^2)\ln^2(x^2 + y^2)}$

• Partial derivative with respect to $y$: $z_y = \frac{\partial}{\partial y}\left(\frac{1}{\ln(x^2 + y^2)}\right) = -\frac{2y}{(x^2 + y^2)\ln^2(x^2 + y^2)}$

7. $u = e^{x/z} + e^{y/z} + e^{z/x}$

• Show $xu_x + yu_y + zu_z = 0$:
  • Apply Euler’s homogeneous function theorem, assuming $u$ is homogeneous of degree zero.

8. $z = e^{-y}\cos(x-y)$

• Partial derivative with respect to $x$: $z_x = -e^{-y}\sin(x-y)$

• Partial derivative with respect to $y$: $z_y = -e^{-y}\cos(x-y) + e^{-y}\sin(x-y)$

• Show that $z_x + z_y + z = 0$:

9. $z = e^y\cos x + e^x\cos y$

• Second partial derivative with respect to $x$: $\frac{\partial^2 z}{\partial x^2} = -e^y\cos x + e^x\cos y$

• Second partial derivative with respect to $y$: $\frac{\partial^2 z}{\partial y^2} = e^y\cos x - e^x\cos y$

10. $z = \sin^{-1}\left(\frac{x}{y}\right)$

• Apply implicit differentiation method:
• Second partial derivatives can be complex but typically follow implicit differentiation rules.

11. $u = (x^3 - y^3)^2$

• Mixed partial derivative $\frac{\partial^3 u}{\partial x^2 \partial y}$:

• First, find the first and second partial derivative with respect to $x$, then with respect to $y$.

• Mixed partial derivative $\frac{\partial^3 u}{\partial y^2 \partial x}$:

• Similarly, find the derivatives in the indicated order using chain and power rules.

These partial derivatives and evaluations involve applying basic differentiation rules, such as the product, quotient, and chain rules.