Calculadora de Derivada Parcial
Calcular derivadas parciais passo a passo
Esta calculadora online calculará a derivada parcial da função, com as etapas mostradas. Você pode especificar qualquer ordem de integração.
Solution
Your input: find $$$\frac{\partial}{\partial x}\left(2 x^{2} y - 2 x^{2} + y^{3} - 2 y^{2} + 2\right)$$$
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}{\frac{\partial}{\partial x}\left(2 x^{2} y - 2 x^{2} + y^{3} - 2 y^{2} + 2\right)}}={\color{red}{\left(\frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 x^{2}\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right) + \frac{\partial}{\partial x}\left(2 x^{2} y\right)\right)}}$$Apply the constant multiple rule $$$\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$$$ with $$$c=2 y$$$ and $$$f=x^{2}$$$:
$${\color{red}{\frac{\partial}{\partial x}\left(2 x^{2} y\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 x^{2}\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)={\color{red}{2 y \frac{\partial}{\partial x}\left(x^{2}\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 x^{2}\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=2$$$:
$$2 y {\color{red}{\frac{\partial}{\partial x}\left(x^{2}\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 x^{2}\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)=2 y {\color{red}{\left(2 x^{-1 + 2}\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 x^{2}\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)=4 x y + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 x^{2}\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)$$Apply the constant multiple rule $$$\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$$$ with $$$c=2$$$ and $$$f=x^{2}$$$:
$$4 x y - {\color{red}{\frac{\partial}{\partial x}\left(2 x^{2}\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)=4 x y - {\color{red}{\left(2 \frac{\partial}{\partial x}\left(x^{2}\right)\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=2$$$:
$$4 x y - 2 {\color{red}{\frac{\partial}{\partial x}\left(x^{2}\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)=4 x y - 2 {\color{red}{\left(2 x^{-1 + 2}\right)}} + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)=4 x y - 4 x + \frac{\partial}{\partial x}\left(2\right) - \frac{\partial}{\partial x}\left(2 y^{2}\right) + \frac{\partial}{\partial x}\left(y^{3}\right)$$The derivative of a constant is 0:
$$4 x y - 4 x - {\color{red}{\frac{\partial}{\partial x}\left(2 y^{2}\right)}} + \frac{\partial}{\partial x}\left(2\right) + \frac{\partial}{\partial x}\left(y^{3}\right)=4 x y - 4 x - {\color{red}{\left(0\right)}} + \frac{\partial}{\partial x}\left(2\right) + \frac{\partial}{\partial x}\left(y^{3}\right)$$The derivative of a constant is 0:
$$4 x y - 4 x + {\color{red}{\frac{\partial}{\partial x}\left(2\right)}} + \frac{\partial}{\partial x}\left(y^{3}\right)=4 x y - 4 x + {\color{red}{\left(0\right)}} + \frac{\partial}{\partial x}\left(y^{3}\right)$$The derivative of a constant is 0:
$$4 x y - 4 x + {\color{red}{\frac{\partial}{\partial x}\left(y^{3}\right)}}=4 x y - 4 x + {\color{red}{\left(0\right)}}=4 x \left(y - 1\right)$$Thus, $$$\frac{\partial}{\partial x}\left(2 x^{2} y - 2 x^{2} + y^{3} - 2 y^{2} + 2\right)=4 x \left(y - 1\right)$$$
Answer: $$$\frac{\partial}{\partial x}\left(2 x^{2} y - 2 x^{2} + y^{3} - 2 y^{2} + 2\right)=4 x \left(y - 1\right)$$$