# 2変数関数のテイラー（Taylor）の定理の導出

(1)式

${\int }_{a}^{a+h}\frac{\partial }{\partial x}{f}_{y}\left(x,b\right)dx$

$={\int }_{a}^{a+h}1\cdot {f}_{yx}\left(x,b\right)dx$

$={\int }_{a}^{a+h}-{\left(a+h-x\right)}^{\prime }\cdot {f}_{yx}\left(x,b\right)dx$

$={\left[-\left(a+h-x\right){f}_{yx}\left(x,b\right)\right]}_{a}^{a+h}$$-\underset{a}{\overset{a+h}{\int }}-\left(a+h-x\right)\frac{\partial }{\partial x}{f}_{yx}\left(x,b\right)dx$

$=h{f}_{yx}\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}\left(a+h-x\right){f}_{yxx}\left(x,b\right)dx$

$=h{f}_{yx}\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}}^{\prime }{f}_{yxx}\left(x,b\right)dx$

$=h{f}_{yx}\left(a,b\right)$$+{\left[-\frac{1}{2}{\left(a+h-x\right)}^{2}{f}_{yxx}\left(x,b\right)\right]}_{a}^{a+h}$ $-\underset{a}{\overset{a+h}{\int }}\left\{-\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}\frac{\partial }{\partial x}{f}_{yxx}\left(x,b\right)dx$

$=h{f}_{yx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yxx}\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}\left\{\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}{f}_{yxxx}\left(x,b\right)dx$

$=h{f}_{yx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yxx}\left(a,b\right)$$+\cdots$$+\frac{1}{\left(n-1\right)!}{h}^{n-1}{\left(\frac{\partial }{\partial x}\right)}^{n-1}\frac{\partial }{\partial y}f\left(a,b\right)$

$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{n!}{\left(a+h-x\right)}^{n}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{n}\frac{\partial }{\partial y}f\left(x,b\right)dx$

(2)式

${\int }_{a}^{a+h}\frac{\partial }{\partial x}{f}_{yy}\left(x,b\right)dx$

$={\int }_{a}^{a+h}1\cdot {f}_{yyx}\left(x,b\right)dx$

$={\int }_{a}^{a+h}-{\left(a+h-x\right)}^{\prime }\cdot {f}_{yyx}\left(x,b\right)dx$

$={\left[-\left(a+h-x\right){f}_{yyx}\left(x,b\right)\right]}_{a}^{a+h}$$-\underset{a}{\overset{a+h}{\int }}-\left(a+h-x\right)\frac{\partial }{\partial x}{f}_{yyx}\left(x,b\right)dx$

$=h{f}_{yyx}\left(a,b\right)+{\int }_{a}^{a+h}\left(a+h-x\right){f}_{yyxx}\left(x,b\right)dx$

$=h{f}_{yyx}\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}}^{\prime }{f}_{yyxx}\left(x,b\right)dx$

$=h{f}_{yyx}\left(a,b\right)$$+{\left[-\frac{1}{2}{\left(a+h-x\right)}^{2}{f}_{yyxx}\left(x,b\right)\right]}_{a}^{a+h}$$-\underset{a}{\overset{a+h}{\int }}\left\{-\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}\frac{\partial }{\partial x}{f}_{yyxx}\left(x,b\right)dx$

$=h{f}_{yyx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yyxx}\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}\left\{\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}{f}_{yyxxx}\left(x,b\right)dx$

(1)式と同様に，積分を繰り返すと

$=h{f}_{yyx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yyxx}\left(a,b\right)$$+\cdots$$+\frac{1}{\left(n-2\right)!}{h}^{n-2}{\left(\frac{\partial }{\partial x}\right)}^{n-2}{\left(\frac{\partial }{\partial y}\right)}^{2}f\left(a,b\right)$

$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{\left(n-1\right)!}{\left(a+h-x\right)}^{n-1}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{n-1}{\left(\frac{\partial }{\partial y}\right)}^{2}f\left(x,b\right)dx$

(3)式

${\int }_{a}^{a+h}\frac{\partial }{\partial x}{f}_{yyy}\left(x,b\right)dx$

$={\int }_{a}^{a+h}1\cdot {f}_{yyyx}\left(x,b\right)dx$

$={\int }_{a}^{a+h}-{\left(a+h-x\right)}^{\prime }{f}_{yyyx}\left(x,b\right)dx$

$={\left[-\left(a+h-x\right){f}_{yyyx}\left(x,b\right)\right]}_{a}^{a+h}$$-\underset{a}{\overset{a+h}{\int }}-\left(a+h-x\right)\frac{\partial }{\partial x}{f}_{yyyx}\left(x,b\right)dx$

$=h{f}_{yyyx}\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}}^{\prime }{f}_{yyyxx}\left(x,b\right)dx$

$=h{f}_{yyyx}\left(a,b\right)$$+{\left[-\frac{1}{2}{\left(a+h-x\right)}^{2}{f}_{yyyxx}\left(x,b\right)\right]}_{a}^{a+h}$$-\underset{a}{\overset{a+h}{\int }}\left\{-\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}\frac{\partial }{\partial x}{f}_{yyyxx}\left(x,b\right)dx$

$=h{f}_{yyyx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yyyxx}\left(a,b\right)$$+\underset{a}{\overset{a+h}{\int }}\left\{\frac{1}{2}{\left(a+h-x\right)}^{2}\right\}{f}_{yyyxxx}\left(x,b\right)dx$

$=h{f}_{yyyx}\left(a,b\right)+\frac{1}{2}{h}^{2}{f}_{yyyxx}\left(a,b\right)$$+\cdots$$+\frac{1}{\left(n-3\right)!}{h}^{n-3}{\left(\frac{\partial }{\partial x}\right)}^{n-3}{\left(\frac{\partial }{\partial y}\right)}^{3}f\left(a,b\right)$

$+\underset{a}{\overset{a+h}{\int }}{\left\{-\frac{1}{\left(n-2\right)!}{\left(a+h-x\right)}^{n-2}\right\}}^{\prime }{\left(\frac{\partial }{\partial x}\right)}^{n-2}{\left(\frac{\partial }{\partial y}\right)}^{3}f\left(x,b\right)dx$

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