Calculadora de Derivadas Parciais

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

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

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

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

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

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

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