📝 题目
6．设 $f$ 具有一阶连续偏导数，求下列函数的一阶偏导数． （1）$z=f(3 x+2 y, 4 x-3 y)$ ； （2）$z=f\left(x^{2}-y^{2}, \mathrm{e}^{x y}\right)$ ； （3）$z=f(y \ln x, 2 x+3 y)$ ； （4）$z=f\left(\frac{y}{x}, \frac{x}{y}\right)$ ； （5）$z=f(x, x+y, x-y)$ ； （6）$u=f(x, x y, x y z)$ ．
💡 答案与解析
[AI解答]
[AI解答] 以下为各小题的一阶偏导数求解过程，其中 $f$ 具有一阶连续偏导数，记 $f$ 对第 $i$ 个中间变量的偏导数为 $f_i'$。
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### （1）$z = f(3x+2y,\, 4x-3y)$
设中间变量： $u = 3x+2y,\quad v = 4x-3y$ 则 $$ \frac{\partial z}{\partial x} = f_1' \cdot \frac{\partial u}{\partial x} + f_2' \cdot \frac{\partial v}{\partial x} = f_1' \cdot 3 + f_2' \cdot 4 $$ $$ \frac{\partial z}{\partial y} = f_1' \cdot \frac{\partial u}{\partial y} + f_2' \cdot \frac{\partial v}{\partial y} = f_1' \cdot 2 + f_2' \cdot (-3) $$
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### （2）$z = f(x^2 - y^2,\, e^{xy})$
设 $u = x^2 - y^2,\quad v = e^{xy}$ $$ \frac{\partial z}{\partial x} = f_1' \cdot 2x + f_2' \cdot y e^{xy} $$ $$ \frac{\partial z}{\partial y} = f_1' \cdot (-2y) + f_2' \cdot x e^{xy} $$
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### （3）$z = f(y \ln x,\, 2x+3y)$
设 $u = y \ln x,\quad v = 2x+3y$ $$ \frac{\partial z}{\partial x} = f_1' \cdot \frac{y}{x} + f_2' \cdot 2 $$ $$ \frac{\partial z}{\partial y} = f_1' \cdot \ln x + f_2' \cdot 3 $$
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### （4）$z = f\left(\frac{y}{x},\, \frac{x}{y}\right)$
设 $u = \frac{y}{x},\quad v = \frac{x}{y}$ $$ \frac{\partial z}{\partial x} = f_1' \cdot \left(-\frac{y}{x^2}\right) + f_2' \cdot \frac{1}{y} $$ $$ \frac{\partial z}{\partial y} = f_1' \cdot \frac{1}{x} + f_2' \cdot \left(-\frac{x}{y^2}\right) $$
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### （5）$z = f(x,\, x+y,\, x-y)$
设 $u = x,\quad v = x+y,\quad w = x-y$ $$ \frac{\partial z}{\partial x} = f_1' \cdot 1 + f_2' \cdot 1 + f_3' \cdot 1 $$ $$ \frac{\partial z}{\partial y} = f_1' \cdot 0 + f_2' \cdot 1 + f_3' \cdot (-1) = f_2' - f_3' $$
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### （6）$u = f(x,\, xy,\, xyz)$
设 $u_1 = x,\quad u_2 = xy,\quad u_3 = xyz$ $$ \frac{\partial u}{\partial x} = f_1' \cdot 1 + f_2' \cdot y + f_3' \cdot yz $$ $$ \frac{\partial u}{\partial y} = f_1' \cdot 0 + f_2' \cdot x + f_3' \cdot xz $$ $$ \frac{\partial u}{\partial z} = f_1' \cdot 0 + f_2' \cdot 0 + f_3' \cdot xy $$
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**难度评级**：★★☆☆☆ （主要考察链式法则的直接应用，计算量小，但需注意中间变量个数及求导顺序）