若f0（x）=sinx，f1（x）=f0′（x），f2（x）=f1′（x），…，fn+1（x）=fn′（x），n∈N，则f2009′（x）=______．
由题意f0（x）=sinx，f1（x）=f0′（x）=cosx，f2（x）=f1′（x）=-sinx，f3（x）=f2′（x）=-cosx，f4（x）=f3′（x）=sinx，由此可知，在逐次求导的过程中，所得的函数呈周期性变化，从0开始计，周期是4，
∵2009=4×502+1，f2009（x）是一周中的第一个函数，故f2009（x）=cosx
故答案为cosx
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| One summer in college, I was invited to be an instructor at a high school leadership camp. I first 1 a boy under the tree on the first day of camp. His obvious 2 and shyness made him appear weak and lonely. Nearby, 200 13 campers were playing and joking, but the boy seemed to want to be anywhere 4 where he was. I was instructed to care more about campers who might feel 5 . So I 6 him and said, "Hi, I'm Kevin. It's nice to meet you. How are you?" 7 a shaky voice he 8 answered, "Okay, I guess." I calmly asked him to join in the activities and 9 some new people. He quietly replied, "No, this is not really my thing." I could 10 that this whole experience was 11 to him. But I somehow knew it wouldn't be right to 12 him, either. It was going to take more time and 13 . The next day, I was leading camp songs for the campers. They eagerly participated. But the boy was just sitting alone, 14 out the window. That evening at our nightly staff meeting, I made my 15 about him known. I asked them to pay special attention and spend time with him 16 they could. The days flew by fast. When the "last dance" came, surprisingly, the boy from under the tree was now a shirtless dancing 17 . He owned the dance floor 18 meaningful time with others. I couldn't believe it was him. In that instant, I realized how easy it is to give a bit of 19 every day. You may never know how much each gesture may mean to someone else. I tell this story as 20 as I can, and I advise others to look out for their own "boy under the tree." | ||||
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