$BB?JQ?tHG(B Taylor $B$NDjM}(B

Proof. ($B%M%;%O%c!"%?!"%"%;%l%U%?!"%-!"!+!"%1!#!W(B) $m$ $B!"%R%(%j!"%1!"k5"G $B$KBP$7$F$O!"(B chain rule $B$+$i(B

$$F^{(1)}(t) =F'(t)=\frac{\D}{\D t}f(a+th) =f'(a+th)\frac{\D}{\D t}(a+ th)=f'(a+th)h$$

$$=\sum_{i=1}^n\frac{\rd f}{\rd x_i}(a+th)h_i.$$

$B$3$l$O(B $\dsp\sum_{1\le i_1\le n} \frac{\rd f}{\rd x_{i_1}}(a+th)h_{i_1}$ $B$H=q$-D>$;$k!#(B $B$f$($K(B $m=1$ $B$N$H$-@.N)$9$k!#(B

$m$ $B$N$H$-@.N)$9$k!"$9$J$o$A(B

$$F^{(m)}(t) =\sum_{1\le i_1,i_2,\dots,i_m\le n} \frac{\rd^m f}{\rd x_{i_1}\rd x_{i_2}\cdots\rd x_{i_m}} (a+t h)h_{i_1}h_{i_2}\cdots h_{i_m}$$

$B$H2>Dj$9$k$H!"(B

$$F^{(m+1)}(t) =\frac{\D}{\D t}F^{(m)}(t) =\sum_{1\le i_1,i_2,\dots,i_m\le n} \f... ...f}{\rd x_{i_1}\rd x_{i_2}\cdots\rd x_{i_m}} (a+t h)h_{i_1}h_{i_2}\cdots h_{i_m}$$

$$=\sum_{1\le i_1,i_2,\dots,i_m\le n} \left( \sum_{i=1}^n \frac{\rd... ...1}\rd x_{i_2}\cdots\rd x_{i_m}} (a+t h)h_i \right) h_{i_1}h_{i_2}\cdots h_{i_m}$$

$$=\sum_{1\le i_1,i_2,\dots,i_m,i_{m+1}\le n} \frac{\rd^{m+1} f}{\r... ...dots\rd x_{i_m}\rd x_{i_{m+1}}} (a+t h)h_{i_1}h_{i_2}\cdots h_{i_m}h_{i_{m+1}}.$$

$B$3$l$O(B $m+1$ $B$N$H$-$b@.N)$9$k$3$H$r<($9!#(B $B$f$($K$9$Y$F$N(B $m(\le k)$ $B$K$D$$$F@.N)$9$k!#(B $\qedsymbol$ ARRAY(0xf4d904) $\qedsymbol$

$B5-9f$NLsB+(B: $f$ $B$N(B $x$ $B$K$*$1$k(B $m$ $B

$$(d^m f)_x(h) :=\sum_{1\le i_1,i_2,\dots,i_m\le n} \frac{\rd^m f}{\rd x_{i_1}\rd x_{i_2}\cdots\rd x_{i_m}} (x)h_{i_1}h_{i_2}\cdots h_{i_m}$$

$$=\sum_{\text{$\alpha_1,\dots,\alpha_n$\ $B$OHsIi@0?t(B}\atop \alpha_1... ...\cdots\rd x_n^{\alpha_n}} (x)h_1^{\alpha_1}h_2^{\alpha_2}\cdots h_n^{\alpha_n}.$$

$B$3$l$r;H$&$H!"(B

$$F^{(m)}(t)=(d^m f)_x(h),\quad F^{(m)}(0)=(d^m f)_a(h).$$