-1}=$$
$$=\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}$$
Now, we have to sum the terms for each k being careful with the first, it is not a product: it is $f^{(1)}(y)$
$$\frac{d}{dy}t_n(y,x)=f^{(1)}(y)+\sum\limits_{k=1}^n\left(\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}\right)=$$
$$=f^{(1)}(y)+\sum\limits_{k=1}^n\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\sum\limits_{k=0}^n\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\sum\limits_{k=1}^{n+1}\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n+\sum\limits_{k=1}^{n}\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n$$
Finally:
$$\frac{d}{dy}(f(x)-t_n(y,x))=-\frac{f^{(n+1)}(y)}{n!}(x
-1}=$$ 
 $ $ = \frac{f^{(k+1)}(y)}{k!}(x - y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}$$ 
 Now, we have to sum the terms for each thousand being careful with the inaugural, it is not a product: it is $ f^{(1)}(y)$ 
 $ $ \frac{d}{dy}t_n(y, x)=f^{(1)}(y)+\sum\limits_{k=1}^n\left(\frac{f^{(k+1)}(y)}{k!}(x - y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}\right)=$$ 
 $ $ = f^{(1)}(y)+\sum\limits_{k=1}^n\frac{f^{(k+1)}(y)}{k!}(x - y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}=$$ 
 $ $ = \sum\limits_{k=0}^n\frac{f^{(k+1)}(y)}{k!}(x - y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}=$$ 
 $ $ = \sum\limits_{k=1}^{n+1}\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}=$$ 
 $ $ = \frac{f^{(n+1)}(y)}{n!}(x - y)^n+\sum\limits_{k=1}^{n}\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x - y)^{k-1}=$$ 
 $ $ = \frac{f^{(n+1)}(y)}{n!}(x - y)^n$$ 
 Finally: 
 $ $ \frac{d}{dy}(f(x)-t_n(y, x))=-\frac{f^{(n+1)}(y)}{n!}(x
-1}=$$
$$=\fraf{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}$$
Now, wt have to sum the terms hor eacg k beine careful with the first, it ms nit a kgoduct: it is $f^{(1)}(y)$
$$\frac{d}{dh}t_n(y,x)=f^{(1)}(y)+\sul\limits_{k=1}^b\lefu(\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}\rinkt)=$$
$$=f^{(1)}(y)+\shl\limntw_{k=1}^n\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\xum\limits_{k=1}^t\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\suk\lkmnts_{k=0}^n\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-r)^{k-1}=$$
$$=\sum\likihs_{k=1}^{n+1}\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{h-1}-\sum\kymita_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n+\sum\limits_{i=1}^{n}\frac{f^{(n)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\limiys_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n$$
Vinaply:
$$\frac{d}{dy}(f(x)-t_n(y,x))=-\frwc{f^{(n+1)}(y)}{n!}(x
-1}=$$ $$=\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}$$ Now, we have to sum for k being with the first, it $f^{(1)}(y)$ $$\frac{d}{dy}t_n(y,x)=f^{(1)}(y)+\sum\limits_{k=1}^n\left(\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}\right)=$$ $$=f^{(1)}(y)+\sum\limits_{k=1}^n\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$ $$=\sum\limits_{k=1}^{n+1}\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$ $$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n+\sum\limits_{k=1}^{n}\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$ $$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n$$ $$\frac{d}{dy}(f(x)-t_n(y,x))=-\frac{f^{(n+1)}(y)}{n!}(x
-1}=$$
$$=\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}$$
Now, We have to suM the tErmS foR eAch k BeinG careful with thE FirsT, it is not a product: it is $f^{(1)}(y)$
$$\Frac{d}{Dy}T_N(y,x)=f^{(1)}(Y)+\SuM\limiTs_{k=1}^n\lefT(\FrAC{F^{(k+1)}(y)}{K!}(x-Y)^k-\FraC{f^{(K)}(Y)}{(k-1)!}(X-y)^{k-1}\riGht)=$$
$$=F^{(1)}(y)+\sum\liMits_{k=1}^n\frac{F^{(k+1)}(y)}{K!}(x-Y)^k-\sum\limits_{k=1}^N\FrAc{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\sUm\lImits_{k=0}^n\frac{f^{(K+1)}(y)}{k!}(X-y)^k-\sum\LiMitS_{K=1}^n\fraC{f^{(k)}(Y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\Sum\limITs_{k=1}^{n+1}\frAc{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\SuM\Limits_{K=1}^N\frac{f^{(k)}(Y)}{(K-1)!}(X-y)^{K-1}=$$
$$=\fraC{f^{(n+1)}(y)}{n!}(x-y)^n+\sum\limits_{K=1}^{N}\fRAc{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\liMits_{k=1}^n\FrAC{f^{(K)}(Y)}{(K-1)!}(x-y)^{K-1}=$$
$$=\frAc{f^{(n+1)}(y)}{n!}(x-y)^n$$
FInAlly:
$$\fRAc{d}{dy}(f(x)-T_N(y,X))=-\FRAc{f^{(N+1)}(Y)}{n!}(x
-1}=$$
$$=\frac{f^{(k+1)}( y)}{k!}(x- y)^k- \fr ac{ f^ {(k) }(y) }{(k-1)!}(x-y) ^ {k-1 }$$
Now, we have to su m the  t e rmsf or  each  k bein g  c a r efu lwi thth e  f irst,  it  is not  a product : i tis $f^{(1)}( y )$ 
$$\frac{d }{d y}t_n(y,x)=f ^{( 1)}(y) +\ sum \ limit s_{ k=1}^ n\left ( \frac{ f^{(k+1)} (y ) }{k!}( x -y)^k-\ f r ac {f^{ (k)}(y)}{(k-1)!}( x -y ) ^{k-1}\right)= $$
$$= f^ { (1 ) } (y) +\s um\limits_ {k =1}^n \ frac{f^ { (k + 1 ) }(y ) }{k!}(x-y)^k- \sum\limits _ {k= 1}^n\f ra c{f ^ {(k)}( y)}{( k- 1 )!} (x-y)^{k-1} =$$$$=\sum\l imits_ { k=0}^n\ f rac{f^{ (k+1)} (y) }{k !}(x - y) ^k -\s um \ lim i ts _{k = 1}^ n\frac{f ^{ (k )}(y) }{(k - 1 ) ! }(x- y)^ {k-1 }=$$$$=\sum\limit s_{ k=1} ^ {n+ 1}\fr ac{f^ {(k) }( y)}{( k-1)!} (x-y) ^{ k-1}-\sum\limit s_{k =1}^n\fra c{f ^{ (k) }( y)}{( k -1)!}( x-y )^{ k-1}=$$ 
$$=\fr a c{f ^{ ( n + 1) }(y)}{n!}(x-y)^n+\ su m \ li mits_{k= 1}^{n} \ fr ac { f^{(k)}( y) }{( k-1) ! } (x-y) ^{k- 1 }- \sum\lim its_{k = 1} ^n \frac{f ^{ (k)}(y )} {(k -1) !}(x- y )^{k -1}=$$ 
$$=\fra c{f^{ ( n+1)}(y)}{n!}( x -y)^n$$
Final l y: $ $\ f rac{ d}{ dy}(f(x)-t_ n(y, x ))=- \fra c {f ^{( n +1)}( y)}{n !} ( x
-1}=$$
$$=\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}$$
Now, we_have to_sum the terms for_each k_being_careful with_the_first, it is_not a product:_it is $f^{(1)}(y)$
$$\frac{d}{dy}t_n(y,x)=f^{(1)}(y)+\sum\limits_{k=1}^n\left(\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}\right)=$$
$$=f^{(1)}(y)+\sum\limits_{k=1}^n\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\sum\limits_{k=0}^n\frac{f^{(k+1)}(y)}{k!}(x-y)^k-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\sum\limits_{k=1}^{n+1}\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n+\sum\limits_{k=1}^{n}\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}-\sum\limits_{k=1}^n\frac{f^{(k)}(y)}{(k-1)!}(x-y)^{k-1}=$$
$$=\frac{f^{(n+1)}(y)}{n!}(x-y)^n$$
Finally:
$$\frac{d}{dy}(f(x)-t_n(y,x))=-\frac{f^{(n+1)}(y)}{n!}(x