2-) Findallfirst-orderpartialderivatives of thefollowingfunctions.
a) \( f(x, y)=\left(x^{2}-y^{3}\right)^{4} \)
b) \( f(x, y)=\frac{x}{y}-\frac{y}{x} \)
c) \( f(x, y, z)=e^{x y} \ln (x z) \)
Answer
\( \frac{\partial f(x, y, z)}{\partial z} =e^{xy}\frac{1}{z} \)
Step 1 ：\( \frac{\partial f(x, y)}{\partial x} =4\left(x^{2}-y^{3}\right)^{3}(2x) \)
Step 2 ：\( \frac{\partial f(x, y)}{\partial y} =4\left(x^{2}-y^{3}\right)^{3}(-3y^{2}) \)
Step 3 ：\( \frac{\partial f(x, y)}{\partial x} =\frac{1}{y} + \frac{y}{x^{2}} \)
Step 4 ：\( \frac{\partial f(x, y)}{\partial y} =-\frac{x}{y^{2}} - \frac{1}{x} \)
Step 5 ：\( \frac{\partial f(x, y, z)}{\partial x} =ye^{xy}\ln(xz)+\frac{e^{xy}}{x}\ln(xz) \)
Step 6 ：\( \frac{\partial f(x, y, z)}{\partial y} =xe^{xy}\ln(xz) \)
Step 7 ：\( \frac{\partial f(x, y, z)}{\partial z} =e^{xy}\frac{1}{z} \)
