Υπολογιστής μερικών παραγώγων

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

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

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

Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=2$$$:

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

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

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