1. The document discusses the concept of derivatives and average rate of change in calculus. It provides examples of taking the derivative of several common functions like x^n, sin(x), cos(x), and tan(x). 2. Examples are given of calculating the derivative of each function by taking the limit as h approaches 0 of the difference quotient (f(x+h)-f(x))/h. 3. For each example, the document shows that the derivative is equal to the known formula such as nx^n-1 for f(x)=x^n, cos(x) for f(x)=sin(x), etc.

1. PHYSIXMUS 4th

2. 2010. 3. 6 SAT 잘게 나누어 관찰하기 2 1. 미분일반

4. 평균변화율 f(x2)-f(x1) f(x) x2-x1 f(x2) f(x) f(x) x f(x1) 평균속도 x 0 X x2 x1

5. 변화율 Rate of change f(x) x2-x1->0 Q f(x1) f(x) ->? x P F(x2) F’(x1) x x2 x1

6. 기울기 Gradient

8. 도함수derivative f’: x f’(x)

9. 사례1 EXAMPLE1 f(x)=xn f’(x)=nxn-1

10. f(x+h)-f(x) h->0 ? h ? (x+h)n-xn h->0 h (nC0xn+nC1xn-1h+nC2xn-2h2+…+nCn-1xhn-1+nCnhn)-xn h nC1xn-1+nC2xn-2h+…+nCn-1xhn-2+ nCnhn-1 h->0 nxn-1

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13. 사례2 EXAMPLE2 f(x)=sin x f’(x)=cosx

14. f(x+h)-f(x) h->0 ? h ? sin(x+h)-sinx h->0 h COMPUTER cos x

15. 사례3 EXAMPLE3 f(x)=cos x f’(x)=-sin x

16. f(x+h)-f(x) h->0 ? h ? cos(x+h)-cosx h->0 h COMPUTER -sin x

17. 사례4 EXAMPLE4 f(x)=tan x f’(x)=sec2 x

18. f(x+h)-f(x) h->0 ? h ? tan(x+h)-tanx h->0 h COMPUTER sec2 x

20. 2010. 3. 13 SAT 잘게 나누어 관찰하기 3 1. 여러가지 미분 (Computer)