Partial Derivatives Example 6 Solution

$$\frac{\partial}{\partial x}\left(e^{2x+y^3}\right) = e^{2x+y^3}\frac{\partial}{\partial x}(2x+y^3) = e^{2x+y^3}\left(2\frac{\partial}{\partial x}x+\frac{\partial}{\partial x}y^3\right) = e^{2x+y^3}(2(1)+0)=2e^{2x+y^3}$$

$$\frac{\partial}{\partial y}\left(e^{2x+y^3}\right) = e^{2x+y^3}\frac{\partial}{\partial y}(2x+y^3) = e^{2x+y^3}\left(2\frac{\partial}{\partial y}x+\frac{\partial}{\partial y}y^3\right) = e^{2x+y^3}(2(0)+3y^2)=3y^2e^{2x+y^3}$$