Partial Derivatives

1. Find the partial derivative as a limit to calculate \(\displaystyle{\frac{\partial z}{\partial x}} \) and \( \displaystyle{\frac{\partial z}{\partial y}}\) for the function.   solution

\[z = f(x,y) = x^3y+2x^2y^2-5 \]

2. Find \(\displaystyle{\frac{\partial z}{\partial x}} \) and \( \displaystyle{\frac{\partial z}{\partial y}}\) for the following functions:    solution

(a)  \(\displaystyle{z=x^4\,e^{2y}} \)       (b)  \(\displaystyle{z=\ln(x^3+y^4)} \)      (c)  \(\displaystyle{z=\sin(3x-4y)} \)

3. Find \(\displaystyle{\frac{\partial z}{\partial x}} \) and \( \displaystyle{\frac{\partial z}{\partial y}}\) for the following functions:    solution

(a)  \(\displaystyle{z=\frac{x}{y}} \)        (b)   \(\displaystyle{z=x \cos(xy)} \)