合成関数の偏微分

■問題

$3$\displaystyle{z=x\tan y\text{\hspace{1} },\text{\hspace{1} }x={\sin }^{ - 1 }2t\text{\hspace{1} },\text{\hspace{1} }y={\cos }^{ - 1 }2t\text{\hspace{1} }}$ のとき， $3$\displaystyle{\frac{\hspace{2} dz \hspace{2}}{\hspace{2} dt \hspace{2}}\text{\hspace{1} }}$ を求めよ．

■答

$3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} 2{t}^{2}\sqrt{ 1- 4{t}^{2} } \hspace{2}}}$

■ヒント

合成関数の偏微分の公式を用いて偏微分する．

■解説

$3$\displaystyle{{\sin }^{ - 1 }}$ の微分の公式より

$3$\displaystyle{\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}=\frac{\hspace{2} { \left({ 2t }\right) }^{\prime } \hspace{2}}{\hspace{2} \sqrt{ 1- { \left({ 2t }\right) }^{2} } \hspace{2}}=\frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}}$

$3$\displaystyle{{\cos }^{ - 1 }}$ の微分の公式より

$3$\displaystyle{\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}}=- \frac{\hspace{2} { \left({ 2t }\right) }^{\prime } \hspace{2}}{\hspace{2} \sqrt{ 1- { \left({ 2t }\right) }^{2} } \hspace{2}}=- \frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \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{\hspace{1}{z}_{x}=\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}x \hspace{2}}=\tan y}$

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{\hspace{1}{z}_{y}=\frac{\hspace{2} {\partial}z \hspace{2}}{\hspace{2} {\partial}y \hspace{2}}=x\text{\hspace{1} }\cdot \text{\hspace{1} }\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \cos }^{2}y \hspace{2}}=\frac{\hspace{2}x\hspace{2}}{\hspace{2} { \cos }^{2}x \hspace{2}}}$

合成関数の偏微分の公式より

$3$\displaystyle{\frac{\hspace{2} dz \hspace{2}}{\hspace{2} dt \hspace{2}}}$ $3$\displaystyle{=\hspace{1}{z}_{x}\frac{\hspace{2} dx \hspace{2}}{\hspace{2} dt \hspace{2}}+\hspace{1}{z}_{y}\frac{\hspace{2} dy \hspace{2}}{\hspace{2} dt \hspace{2}}}$

上で求めた式をそれぞれ代入すると

$3$\displaystyle{=\tan y\left({ \frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}} }\right)+\frac{\hspace{2}x\hspace{2}}{\hspace{2} { \cos }^{2}y \hspace{2}}\left({ - \frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}} }\right)}$

$3$\displaystyle{=\frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}\left{{ \frac{\hspace{2} \sin \left({ { \cos }^{ - 1 }2t }\right) \hspace{2}}{\hspace{2} \cos \left({ { \cos }^{ - 1 }2t }\right) \hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} { \left({ 2t }\right) }^{2} \hspace{2}} }\right}}$

$3$\displaystyle{=\frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}\left{{ \frac{\hspace{2} \sqrt{ 1- { \cos }^{2}\left({ { \cos }^{ - 1 }2t }\right) } \hspace{2}}{\hspace{2} 2t \hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} 4{t}^{2} \hspace{2}} }\right}}$

$3$\displaystyle{=\frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}\left{{ \frac{\hspace{2} \sqrt{ 1- { \left({ 2t }\right) }^{2} } \hspace{2}}{\hspace{2} 2t \hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} 4{t}^{2} \hspace{2}} }\right}}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} 2{t}^{2}\sqrt{ 1- 4{t}^{2} } \hspace{2}}}$

■別解

$3$\displaystyle{z=x\tan y}$ に $3$\displaystyle{x={\sin }^{ - 1 }2t}$ ， $3$\displaystyle{y={\cos }^{ - 1 }2t}$ を代入する．

$3$\displaystyle{z=\left({ { \sin }^{ - 1 }2t }\right)\tan \left({ { \cos }^{ - 1 }2t }\right)}$

よって

$3$\displaystyle{\frac{\hspace{2} dz \hspace{2}}{\hspace{2} dt \hspace{2}}=\left({ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dt \hspace{2}}{ \sin }^{ - 1 }2t }\right)\tan \left({ { \cos }^{ - 1 }2t }\right)+\left({ { \sin }^{ - 1 }2t }\right)\left{{ \frac{\hspace{2}d\hspace{2}}{\hspace{2} dt \hspace{2}}\tan \left({ { \cos }^{ - 1 }2t }\right) }\right}}$

$3$\displaystyle{=\frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- { \left({ 2t }\right) }^{2} } \hspace{2}}\frac{\hspace{2} { \sin }^{ - 1 }\left({ { \cos }^{ - 1 }2t }\right) \hspace{2}}{\hspace{2} \cos \left({ { \cos }^{ - 1 }2t }\right) \hspace{2}}+\left({ { \sin }^{ - 1 }2t }\right)\frac{\hspace{2}1\hspace{2}}{\hspace{2} { \left{{ \cos \left({ { \cos }^{ - 1 }2t }\right) }\right} }^{2} \hspace{2}}\left({ - \frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- { \left({ 2t }\right) }^{2} } \hspace{2}} }\right)}$

$3$\displaystyle{=\frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}\cdot \frac{\hspace{2} \sqrt{ 1- { \left{{ \cos \left({ { \cos }^{ - 1 }2t }\right) }\right} }^{2} } \hspace{2}}{\hspace{2} 2t \hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} { \left({ 2t }\right) }^{2} \hspace{2}}\cdot \frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}}$

$3$\displaystyle{=\frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}\cdot \frac{\hspace{2} \sqrt{ 1- { \left({ 2t }\right) }^{2} } \hspace{2}}{\hspace{2} 2t \hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} { \left({ 2t }\right) }^{2} \hspace{2}}\cdot \frac{\hspace{2}2\hspace{2}}{\hspace{2} \sqrt{ 1- 4{t}^{2} } \hspace{2}}}$

$3$\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}t\hspace{2}}- \frac{\hspace{2} { \sin }^{ - 1 }2t \hspace{2}}{\hspace{2} 2{t}^{2}\sqrt{ 1- 4{t}^{2} } \hspace{2}}}$

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最終更新日： 2024年6月30日