Calculadora de Derivadas Parciais

Your input: find $$$\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)$$$

Apply the constant multiple rule $$$\frac{\partial}{\partial z} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial z} \left(f \right)$$$ with $$$c=x y^{2}$$$ and $$$f=z^{3}$$$:

$${\color{red}{\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)}}={\color{red}{x y^{2} \frac{\partial}{\partial z}\left(z^{3}\right)}}$$

Apply the power rule $$$\frac{\partial}{\partial z} \left(z^{n} \right)=n\cdot z^{-1+n}$$$ with $$$n=3$$$:

$$x y^{2} {\color{red}{\frac{\partial}{\partial z}\left(z^{3}\right)}}=x y^{2} {\color{red}{\left(3 z^{-1 + 3}\right)}}=3 x y^{2} z^{2}$$

Thus, $$$\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)=3 x y^{2} z^{2}$$$

Answer: $$$\frac{\partial}{\partial z}\left(x y^{2} z^{3}\right)=3 x y^{2} z^{2}$$$