偏微分の基礎

■問題

次の関数を偏微分せよ．

$3$\displaystyle{ z={ \cos }^{ - 1 }\frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}} }$

■答

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

$3$\displaystyle{ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}=- \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ {x}^{2}- {y}^{2} } \hspace{2}} }$

■ヒント

$3$\displaystyle{ { \cos }^{ - 1 }x }$ の微分を，合成関数の微分によって行う．
偏導関数の定義を用いて偏微分する．
微分後の式に数値を代入する．

■解説

$3$\displaystyle{ u=\frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}} }$ とおくと，

$3$\displaystyle{ z={ \cos }^{ - 1 }u }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}=- \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- {u}^{2} } \hspace{2}} }$

$3$\displaystyle{ u=\frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}} }$ を代入する．

$3$\displaystyle{ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}=- \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- { $ \frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}} $ }^{2} } \hspace{2}} }$

$3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ 1- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} } \hspace{2}} }$

両辺に $3$\displaystyle{x}$ をかける．

$3$\displaystyle{ =- \frac{\hspace{2} x\cdot 1 \hspace{2}}{\hspace{2} x\cdot \sqrt{ 1- \frac{\hspace{2} {y}^{2} \hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} } \hspace{2}} }$

$3$\displaystyle{ =- \frac{\hspace{2}x\hspace{2}}{\hspace{2} \sqrt{ {x}^{2}- {y}^{2} } \hspace{2}} }$

$3$\displaystyle{ u=\frac{\hspace{2}y\hspace{2}}{\hspace{2}x\hspace{2}} }$ を $3$\displaystyle{x}$ と $3$\displaystyle{y}$ でそれぞれ偏微分する．

$3$\displaystyle{ \frac{\hspace{2} {\partial}u \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}=- \frac{\hspace{2}y\hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} }$

$3$\displaystyle{ \frac{\hspace{2} {\partial}u \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}=\frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}} }$

偏導関数の定義より， $3$\displaystyle{y\text{\hspace{1}}}$ を定数とみなして $3$\displaystyle{x\text{\hspace{1}}}$ で微分する．

$3$\displaystyle{ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}=\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\cdot \frac{\hspace{2} {\partial}u \hspace{2}}{\hspace{2} {\partial}x \hspace{2}} }$

$3$\displaystyle{ =- \frac{\hspace{2}x\hspace{2}}{\hspace{2} \sqrt{ {x}^{2}- {y}^{2} } \hspace{2}}\cdot $ - \frac{\hspace{2}y\hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} $ }$

$3$\displaystyle{ =\frac{\hspace{2} xy \hspace{2}}{\hspace{2} {x}^{2}\sqrt{ {x}^{2}- {y}^{2} } \hspace{2}} }$

$3$\displaystyle{ =\frac{\hspace{2}y\hspace{2}}{\hspace{2} x\sqrt{ {x}^{2}- {y}^{2} } \hspace{2}} }$

偏導関数の定義より， $3$\displaystyle{x\text{\hspace{1}}}$ を定数とみなして $3$\displaystyle{y\text{\hspace{1}}}$ で微分する．

$3$\displaystyle{ \frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}=\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}u \hspace{2}}\cdot \frac{\hspace{2} {\partial}u \hspace{2}}{\hspace{2} {\partial}y \hspace{2}} }$

$3$\displaystyle{ =- \frac{\hspace{2}x\hspace{2}}{\hspace{2} \sqrt{ {x}^{2}- {y}^{2} } \hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2}x\hspace{2}} }$

$3$\displaystyle{ =- \frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ {x}^{2}- {y}^{2} } \hspace{2}} }$

ホーム>>カテゴリー分類>>微分>>偏微分>>問題演習>>微分の基礎

学生スタッフ作成

最終更新日： 2023年8月24日