题目
设z=4x^3+3x^2 y+xln(xy)，则z=4x^3+3x^2 y+xln(xy)分别为（ ）z=4x^3+3x^2 y+xln(xy)z=4x^3+3x^2 y+xln(xy)z=4x^3+3x^2 y+xln(xy)z=4x^3+3x^2 y+xln(xy)
设
，则
分别为（ ）
题目解答
答案
解：两边对
求偏导可得，
故本题选
.
解析
步骤 1：计算$\dfrac {\partial z}{\partial x}$
首先，对$z=4{x}^{3}+3{x}^{2}y+x\ln (xy)$关于$x$求偏导数。
$\dfrac {\partial z}{\partial x}=12{x}^{2}+6xy+\ln (xy)+\dfrac {x}{xy}=\dfrac {\partial z}{\partial x}=12{x}^{2}+6xy+\ln (xy)+\dfrac {1}{y}$
步骤 2：计算$\dfrac {{\partial }^{2}z}{\partial {x}^{2}}$
接着，对$\dfrac {\partial z}{\partial x}$关于$x$再次求偏导数。
$\dfrac {{\partial }^{2}z}{\partial {x}^{2}}=24x+6y+\dfrac {1}{y}$
步骤 3：计算$\dfrac {{\partial }^{2}z}{\partial x\partial y}$
最后，对$\dfrac {\partial z}{\partial x}$关于$y$求偏导数。
$\dfrac {{\partial }^{2}z}{\partial x\partial y}=6x+\dfrac {1}{y}$
首先，对$z=4{x}^{3}+3{x}^{2}y+x\ln (xy)$关于$x$求偏导数。
$\dfrac {\partial z}{\partial x}=12{x}^{2}+6xy+\ln (xy)+\dfrac {x}{xy}=\dfrac {\partial z}{\partial x}=12{x}^{2}+6xy+\ln (xy)+\dfrac {1}{y}$
步骤 2：计算$\dfrac {{\partial }^{2}z}{\partial {x}^{2}}$
接着，对$\dfrac {\partial z}{\partial x}$关于$x$再次求偏导数。
$\dfrac {{\partial }^{2}z}{\partial {x}^{2}}=24x+6y+\dfrac {1}{y}$
步骤 3：计算$\dfrac {{\partial }^{2}z}{\partial x\partial y}$
最后，对$\dfrac {\partial z}{\partial x}$关于$y$求偏导数。
$\dfrac {{\partial }^{2}z}{\partial x\partial y}=6x+\dfrac {1}{y}$