# Questions on Partial Derivatives

Find all the Six Partial Derivatives $$f_{x}\; ,\; \; f_{y\; },\; \; f_{xx}\; ,\; f_{yy}\; ,\; \; f_{xy\; },\; \; f_{yx}$$ of the following and verify that $$f_{xy}\; =\; f_{yx}$$

1. $$z\; =\; 8x^{2}\; +\; 4y^{3}\; -\; 10$$
2. $$z\; =\; \frac{x^{2}}{x\; -\; y\; +\; 1}$$
3. $$z\; =\; 2x^{2}\; -\; 11x^{2}y\; +\; 3y^{2}$$
4. $$z\; =\; x^{2}y\; +\; \frac{x^{2}}{y^{3}}\; +\; \frac{y^{2}}{x^{3}}$$
5. $$z\; =\; e^{x^{2}\; +\; 3y^{2}}$$
6. $$z\; =\; x^{4}\; +\; x^{2}y^{2}\; +\; y^{4}$$
7. $$z\; =\; x^{y}$$
8. $$z\; =\; \log \; \frac{x^{2}\; -\; y^{2}}{x^{2}\; +\; y^{2}}$$
9. $$z\; =\; 3x^{2}y\; +\; x^{5}\; +\; 3y^{2}$$
10. $$z\; =\; \frac{x^{2}}{y^{3}}\; +\; \frac{y^{2}}{x^{3}}$$
11. $$z\; =\; x^{2}y\; +\; \frac{x^{2}}{y^{3}}\; +\; \frac{y^{2}}{x^{3}}$$
12. $$z\; =\; 3x^{2}\; +\; 4xy\; +\; 2y^{2}$$
13. $$z\; =\; x^{3}e^{2y}$$
14. $$z\; =\; \log \; \left( x^{2}\; -\; y^{2} \right)$$
15. $$z\; =\; x^{2}\; +\; 2xy\; +\; y^{2}\; +\; 2$$
16. $$z\; =\; x^{2}\; +\; 2hxy\; +\; y^{2}$$
17. $$z\; =\; x^{2}y$$
18. $$z\; =\; x^{2}y\; +\; y^{3}$$
19. $$z\; =\; 2x^{3}\; +\; 5x^{3}y\; +\; xy^{2}\; +\; y^{3}$$
20. $$z\; =\; \left( x^{2}\; +\; y^{2} \right)^{2}$$
21. $$z\; =\; \frac{x\; +\; 4}{2x\; +\; 5y\; }$$
22. $$z\; =\; \log \; \left( \frac{x^{2}\; +\; y^{2}}{xy} \right)$$