関数 $f(x, y) = \sin\left(\frac{5x}{y}\right)$ の4つの2階偏導関数 $f_{xx}(x, y)$、$f_{xy}(x, y)$、$f_{yx}(x, y)$、$f_{yy}(x, y)$ を求める。

1. 問題の内容

関数 $f(x, y) = \sin\left(\frac{5x}{y}\right)$ の4つの2階偏導関数 $f_{xx}(x, y)$、$f_{xy}(x, y)$、$f_{yx}(x, y)$、$f_{yy}(x, y)$ を求める。

2. 解き方の手順

まず、1階偏導関数を求める。

$f_x(x, y) = \frac{\partial}{\partial x} \sin\left(\frac{5x}{y}\right) = \cos\left(\frac{5x}{y}\right) \cdot \frac{5}{y} = \frac{5}{y} \cos\left(\frac{5x}{y}\right)$

$f_y(x, y) = \frac{\partial}{\partial y} \sin\left(\frac{5x}{y}\right) = \cos\left(\frac{5x}{y}\right) \cdot \left(-\frac{5x}{y^2}\right) = -\frac{5x}{y^2} \cos\left(\frac{5x}{y}\right)$

次に、2階偏導関数を求める。

$f_{xx}(x, y) = \frac{\partial}{\partial x} f_x(x, y) = \frac{\partial}{\partial x} \left(\frac{5}{y} \cos\left(\frac{5x}{y}\right)\right) = \frac{5}{y} \cdot \left(-\sin\left(\frac{5x}{y}\right)\right) \cdot \frac{5}{y} = -\frac{25}{y^2} \sin\left(\frac{5x}{y}\right)$

$f_{xy}(x, y) = \frac{\partial}{\partial y} f_x(x, y) = \frac{\partial}{\partial y} \left(\frac{5}{y} \cos\left(\frac{5x}{y}\right)\right) = 5 \left(-\frac{1}{y^2}\cos\left(\frac{5x}{y}\right) + \frac{1}{y} \sin\left(\frac{5x}{y}\right) \cdot \frac{5x}{y^2} \right) = -\frac{5}{y^2}\cos\left(\frac{5x}{y}\right) + \frac{25x}{y^3}\sin\left(\frac{5x}{y}\right)$

$f_{yx}(x, y) = \frac{\partial}{\partial x} f_y(x, y) = \frac{\partial}{\partial x} \left(-\frac{5x}{y^2} \cos\left(\frac{5x}{y}\right)\right) = -\frac{5}{y^2} \cos\left(\frac{5x}{y}\right) - \frac{5x}{y^2} \left(-\sin\left(\frac{5x}{y}\right) \cdot \frac{5}{y}\right) = -\frac{5}{y^2}\cos\left(\frac{5x}{y}\right) + \frac{25x}{y^3}\sin\left(\frac{5x}{y}\right)$

$f_{yy}(x, y) = \frac{\partial}{\partial y} f_y(x, y) = \frac{\partial}{\partial y} \left(-\frac{5x}{y^2} \cos\left(\frac{5x}{y}\right)\right) = -5x \left( \frac{2}{y^3}\cos\left(\frac{5x}{y}\right) + \frac{1}{y^2} \sin\left(\frac{5x}{y}\right) \cdot \frac{5x}{y^2} \right) = -\frac{10x}{y^3}\cos\left(\frac{5x}{y}\right) - \frac{25x^2}{y^4}\sin\left(\frac{5x}{y}\right)$

3. 最終的な答え

$f_{xx}(x, y) = -\frac{25}{y^2} \sin\left(\frac{5x}{y}\right)$

$f_{xy}(x, y) = -\frac{5}{y^2}\cos\left(\frac{5x}{y}\right) + \frac{25x}{y^3}\sin\left(\frac{5x}{y}\right)$

$f_{yx}(x, y) = -\frac{5}{y^2}\cos\left(\frac{5x}{y}\right) + \frac{25x}{y^3}\sin\left(\frac{5x}{y}\right)$

$f_{yy}(x, y) = -\frac{10x}{y^3}\cos\left(\frac{5x}{y}\right) - \frac{25x^2}{y^4}\sin\left(\frac{5x}{y}\right)$