Find all the Six Partial Derivatives \( f_{x}\; ,\; \; f_{y\; },\; \; f_{xx}\; ,\; f_{yy}\; ,\; \; f_{xy\; },\; \; f_{yx} \) of the following and verify that \( f_{xy}\; =\; f_{yx} \)
- \( z\; =\; 8x^{2}\; +\; 4y^{3}\; -\; 10 \)
- \( z\; =\; \frac{x^{2}}{x\; -\; y\; +\; 1} \)
- \( z\; =\; 2x^{2}\; -\; 11x^{2}y\; +\; 3y^{2} \)
- \( z\; =\; x^{2}y\; +\; \frac{x^{2}}{y^{3}}\; +\; \frac{y^{2}}{x^{3}} \)
- \( z\; =\; e^{x^{2}\; +\; 3y^{2}} \)
- \( z\; =\; x^{4}\; +\; x^{2}y^{2}\; +\; y^{4} \)
- \( z\; =\; x^{y} \)
- \( z\; =\; \log \; \frac{x^{2}\; -\; y^{2}}{x^{2}\; +\; y^{2}} \)
- \( z\; =\; 3x^{2}y\; +\; x^{5}\; +\; 3y^{2} \)
- \( z\; =\; \frac{x^{2}}{y^{3}}\; +\; \frac{y^{2}}{x^{3}} \)
- \( z\; =\; x^{2}y\; +\; \frac{x^{2}}{y^{3}}\; +\; \frac{y^{2}}{x^{3}} \)
- \( z\; =\; 3x^{2}\; +\; 4xy\; +\; 2y^{2} \)
- \( z\; =\; x^{3}e^{2y} \)
- \( z\; =\; \log \; \left( x^{2}\; -\; y^{2} \right) \)
- \( z\; =\; x^{2}\; +\; 2xy\; +\; y^{2}\; +\; 2 \)
- \( z\; =\; x^{2}\; +\; 2hxy\; +\; y^{2} \)
- \( z\; =\; x^{2}y \)
- \( z\; =\; x^{2}y\; +\; y^{3} \)
- \( z\; =\; 2x^{3}\; +\; 5x^{3}y\; +\; xy^{2}\; +\; y^{3} \)
- \( z\; =\; \left( x^{2}\; +\; y^{2} \right)^{2} \)
- \( z\; =\; \frac{x\; +\; 4}{2x\; +\; 5y\; } \)
- \( z\; =\; \log \; \left( \frac{x^{2}\; +\; y^{2}}{xy} \right) \)