2o14年高考江苏卷第26题：
已知\(f_0(x)=\dfrac {\sin x}x(x>0).\)，设\(f_n(x)\)为\(f_{n-1}(x)\)的导数，\(n\in{\bf N}^*\)．
(1) 求\(2f_1\left(\dfrac {\pi}2\right)+\dfrac {\pi}2f_2\left(\dfrac {\pi}2\right)\).
(2)证明：\(\forall n\in {\bf N}^*,\left|nf_{n-1}\left(\dfrac {\pi}4\right)+\dfrac {\pi}4f_n\left(\dfrac {\pi}4\right)\right|=\dfrac {\sqrt 2}2\)．
注意到\[xf_0(x)=\sin x,\]两边求导可得\[f_0(x)+xf_1(x)=\cos x.\]进而可得\[2f_1(x)+xf_2(x)=-\sin x,\\3f_2(x)+xf_3(x)=-\cos x,\\\cdots \cdots\\nf_{n-1}(x)+x\cdot f_n(x)=\sin \left(x+\dfrac {\pi}2\cdot n\right).\]
因此不难计算得 (1)的结果为\(-1\)，(2)亦得证．