Partial Derivatives Example 7 Solution

$$\frac{\partial}{\partial x}\big(x\ln(2y)\big) = \ln(2y)\frac{\partial}{\partial x}(x)+x\frac{\partial}{\partial x}(\ln(2y)) = \ln(2y)(1)+x(0)=\ln(2y)$$

$$\frac{\partial}{\partial y}\big(x\ln(2y)\big) = \ln(2y)\frac{\partial}{\partial y}(x)+x\frac{\partial}{\partial y}(\ln(2y)) = \ln(2y)(0)+x\frac{1}{2y}\frac{\partial}{\partial y}(2y) = 0+x\frac{1}{2y}\cdot 2=\frac{x}{y}$$