|
|
|
|
|
|
|
|
|
|
|
|
偏导数计算器
分步计算偏导数
这个在线计算器将计算函数的偏导数，并显示步骤。您可以指定任意积分顺序。
Solution
Your input: find $$$\frac{\partial^{2}}{\partial x \partial y}\left(x^{2} y^{2}\right)$$$
First, find $$$\frac{\partial}{\partial x}\left(x^{2} y^{2}\right)$$$
Apply the constant multiple rule $$$\frac{\partial}{\partial x} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial x} \left(f \right)$$$ with $$$c=y^{2}$$$ and $$$f=x^{2}$$$:
$${\color{red}{\frac{\partial}{\partial x}\left(x^{2} y^{2}\right)}}={\color{red}{y^{2} \frac{\partial}{\partial x}\left(x^{2}\right)}}$$Apply the power rule $$$\frac{\partial}{\partial x} \left(x^{n} \right)=n\cdot x^{-1+n}$$$ with $$$n=2$$$:
$$y^{2} {\color{red}{\frac{\partial}{\partial x}\left(x^{2}\right)}}=y^{2} {\color{red}{\left(2 x^{-1 + 2}\right)}}=2 x y^{2}$$Thus, $$$\frac{\partial}{\partial x}\left(x^{2} y^{2}\right)=2 x y^{2}$$$
Next, $$$\frac{\partial^{2}}{\partial x \partial y}\left(x^{2} y^{2}\right)=\frac{\partial}{\partial y} \left(\frac{\partial}{\partial x}\left(x^{2} y^{2}\right) \right)=\frac{\partial}{\partial y}\left(2 x y^{2}\right)$$$
Apply the constant multiple rule $$$\frac{\partial}{\partial y} \left(c \cdot f \right)=c \cdot \frac{\partial}{\partial y} \left(f \right)$$$ with $$$c=2 x$$$ and $$$f=y^{2}$$$:
$${\color{red}{\frac{\partial}{\partial y}\left(2 x y^{2}\right)}}={\color{red}{2 x \frac{\partial}{\partial y}\left(y^{2}\right)}}$$Apply the power rule $$$\frac{\partial}{\partial y} \left(y^{n} \right)=n\cdot y^{-1+n}$$$ with $$$n=2$$$:
$$2 x {\color{red}{\frac{\partial}{\partial y}\left(y^{2}\right)}}=2 x {\color{red}{\left(2 y^{-1 + 2}\right)}}=4 x y$$Thus, $$$\frac{\partial}{\partial y}\left(2 x y^{2}\right)=4 x y$$$
Therefore, $$$\frac{\partial^{2}}{\partial x \partial y}\left(x^{2} y^{2}\right)=4 x y$$$
Answer: $$$\frac{\partial^{2}}{\partial x \partial y}\left(x^{2} y^{2}\right)=4 x y$$$