1. 次の関数の偏導関数を求めよ。 (1) $x^3 - 2xy + y^2$ (2) $\frac{1}{xy}$ (3) $\frac{x}{y}$ (4) $\frac{1}{x^2 - y^2}$ (5) $\sqrt{2x + 3y}$ (6) $e^{2x+3y}$ (7) $\sin(2x + 3y)$ (8) $\cos(ax^2 + by^2)$ (9) $(x-y) \log(x+y)$

はい、承知しました。数学の問題を解いていきます。

1. 問題の内容

1. 次の関数の偏導関数を求めよ。

(1) $x^3 - 2xy + y^2$

(2) $\frac{1}{xy}$

(3) $\frac{x}{y}$

(4) $\frac{1}{x^2 - y^2}$

(5) $\sqrt{2x + 3y}$

(6) $e^{2x+3y}$

(7) $\sin(2x + 3y)$

(8) $\cos(ax^2 + by^2)$

(9) $(x-y) \log(x+y)$

2. 次の接平面の方程式を求めよ。

(1) $z = x^2 + y^2$ の点 $(1, 1, 2)$ における接平面。

(2) $z = x^2 - y^2$ の点 $(2, 1, 3)$ における接平面。

2. 解き方の手順

1. 偏導関数を求める問題

各関数について、$x$ に関する偏微分 $\frac{\partial z}{\partial x}$ と $y$ に関する偏微分 $\frac{\partial z}{\partial y}$ を計算します。

(1) $z = x^3 - 2xy + y^2$

$\frac{\partial z}{\partial x} = 3x^2 - 2y$

$\frac{\partial z}{\partial y} = -2x + 2y$

(2) $z = \frac{1}{xy} = (xy)^{-1}$

$\frac{\partial z}{\partial x} = -1(xy)^{-2} \cdot y = -\frac{1}{x^2y}$

$\frac{\partial z}{\partial y} = -1(xy)^{-2} \cdot x = -\frac{1}{xy^2}$

(3) $z = \frac{x}{y}$

$\frac{\partial z}{\partial x} = \frac{1}{y}$

$\frac{\partial z}{\partial y} = -\frac{x}{y^2}$

(4) $z = \frac{1}{x^2 - y^2} = (x^2 - y^2)^{-1}$

$\frac{\partial z}{\partial x} = -1(x^2 - y^2)^{-2} \cdot 2x = -\frac{2x}{(x^2 - y^2)^2}$

$\frac{\partial z}{\partial y} = -1(x^2 - y^2)^{-2} \cdot (-2y) = \frac{2y}{(x^2 - y^2)^2}$

(5) $z = \sqrt{2x + 3y} = (2x + 3y)^{\frac{1}{2}}$

$\frac{\partial z}{\partial x} = \frac{1}{2}(2x + 3y)^{-\frac{1}{2}} \cdot 2 = \frac{1}{\sqrt{2x + 3y}}$

$\frac{\partial z}{\partial y} = \frac{1}{2}(2x + 3y)^{-\frac{1}{2}} \cdot 3 = \frac{3}{2\sqrt{2x + 3y}}$

(6) $z = e^{2x+3y}$

$\frac{\partial z}{\partial x} = 2e^{2x+3y}$

$\frac{\partial z}{\partial y} = 3e^{2x+3y}$

(7) $z = \sin(2x + 3y)$

$\frac{\partial z}{\partial x} = 2\cos(2x + 3y)$

$\frac{\partial z}{\partial y} = 3\cos(2x + 3y)$

(8) $z = \cos(ax^2 + by^2)$

$\frac{\partial z}{\partial x} = -2ax\sin(ax^2 + by^2)$

$\frac{\partial z}{\partial y} = -2by\sin(ax^2 + by^2)$

(9) $z = (x - y) \log(x+y)$

$\frac{\partial z}{\partial x} = \log(x+y) + (x-y)\frac{1}{x+y} = \log(x+y) + \frac{x-y}{x+y}$

$\frac{\partial z}{\partial y} = -\log(x+y) + (x-y)\frac{1}{x+y}(-1) = -\log(x+y) - \frac{x-y}{x+y}$

2. 接平面の方程式を求める問題

接平面の方程式は次の式で与えられます。

$z - z_0 = \frac{\partial z}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial z}{\partial y}(x_0, y_0)(y - y_0)$

(1) $z = x^2 + y^2$、点 $(1, 1, 2)$

$\frac{\partial z}{\partial x} = 2x$

$\frac{\partial z}{\partial y} = 2y$

点 $(1, 1)$ における偏微分係数は、

$\frac{\partial z}{\partial x}(1, 1) = 2(1) = 2$

$\frac{\partial z}{\partial y}(1, 1) = 2(1) = 2$

したがって、接平面の方程式は

$z - 2 = 2(x - 1) + 2(y - 1)$

$z - 2 = 2x - 2 + 2y - 2$

$z = 2x + 2y - 2$

(2) $z = x^2 - y^2$、点 $(2, 1, 3)$

$\frac{\partial z}{\partial x} = 2x$

$\frac{\partial z}{\partial y} = -2y$

点 $(2, 1)$ における偏微分係数は、

$\frac{\partial z}{\partial x}(2, 1) = 2(2) = 4$

$\frac{\partial z}{\partial y}(2, 1) = -2(1) = -2$

したがって、接平面の方程式は

$z - 3 = 4(x - 2) - 2(y - 1)$

$z - 3 = 4x - 8 - 2y + 2$

$z = 4x - 2y - 3$

3. 最終的な答え

1. 偏導関数：

(1) $\frac{\partial z}{\partial x} = 3x^2 - 2y$, $\frac{\partial z}{\partial y} = -2x + 2y$

(2) $\frac{\partial z}{\partial x} = -\frac{1}{x^2y}$, $\frac{\partial z}{\partial y} = -\frac{1}{xy^2}$

(3) $\frac{\partial z}{\partial x} = \frac{1}{y}$, $\frac{\partial z}{\partial y} = -\frac{x}{y^2}$

(4) $\frac{\partial z}{\partial x} = -\frac{2x}{(x^2 - y^2)^2}$, $\frac{\partial z}{\partial y} = \frac{2y}{(x^2 - y^2)^2}$

(5) $\frac{\partial z}{\partial x} = \frac{1}{\sqrt{2x + 3y}}$, $\frac{\partial z}{\partial y} = \frac{3}{2\sqrt{2x + 3y}}$

(6) $\frac{\partial z}{\partial x} = 2e^{2x+3y}$, $\frac{\partial z}{\partial y} = 3e^{2x+3y}$

(7) $\frac{\partial z}{\partial x} = 2\cos(2x + 3y)$, $\frac{\partial z}{\partial y} = 3\cos(2x + 3y)$

(8) $\frac{\partial z}{\partial x} = -2ax\sin(ax^2 + by^2)$, $\frac{\partial z}{\partial y} = -2by\sin(ax^2 + by^2)$

(9) $\frac{\partial z}{\partial x} = \log(x+y) + \frac{x-y}{x+y}$, $\frac{\partial z}{\partial y} = -\log(x+y) - \frac{x-y}{x+y}$

2. 接平面の方程式：

(1) $z = 2x + 2y - 2$

(2) $z = 4x - 2y - 3$