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常数和基本初等函数导数公式推导

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常数和基本初等函数导数公式：

1.c′=0

2.(xn)′=nxn−1
(1).(x√)′=12x√
(2).(1x)′=−1x2

3.(sinx)′=cosx

4.(cosx)′=−sinx

5.(lnx)′=1x

6.(logax)′=logaex

7.(ax)′=axlna
(1). (ex)′=ex

8.(tanx)′=sec2x

9.(cotx)′=−csc2x

10.(secx)′=secxtanx

11.(cscx)′=−cscxcotx

12.(arcsinx)′=11−x2√

13.(arccosx)′=−11−x2√

14.(arctanx)′=11+x2

15.(arccotx)′=−11+x2

下面对导数公式进行推导：

(xn)′=nxn−1

由导数的定义：

lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x

得：

(x n)' = lim Δ x \to 0 ( x + Δ x ) 0 0 n - x n Δ x

= lim Δ x \to 0 \sum n m = 1 C m n x m Δ x n - m - x n Δ x

= lim Δ x \to 0 C 0 n x 0 Δ x n + C 1 n x 1 Δ x n - 1 + \dots \dots + C n - 1 n x n - 1 Δ x 1 + C n n x n Δ x 0 - x n Δ x

= lim Δ x \to 0 C 0 n x 0 Δ x n + C 1 n x 1 Δ x n - 1 + \dots \dots + C n - 1 n x n - 1 Δ x 1 Δ x

= n x n - 1

(sinx)′=cosx

由导数的定义：

lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x

得：

(sin x)' = lim Δ x \to 0 sin ( x + Δ x ) - sin x Δ x

= lim Δ x \to 0 2 cos x + Δ x + x 2 sin x + Δ x - x 2 Δ x

= lim Δ x \to 0 2 cos ( x + Δ x 2 ) sin Δ x 2 2 Δ x 2

= lim Δ x \to 0 cos (x + Δ x 2)

= cos x

(cosx)′=−sinx

由导数的定义：

lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x

得：

(cos x)' = lim Δ x \to 0 cos ( x + Δ x ) - cos x Δ x

= lim Δ x \to 0 - 2 sin x + Δ x + x 2 sin x + Δ x - x 2 Δ x

= lim Δ x \to 0 - 2 sin ( x + Δ x 2 ) sin Δ x 2 2 Δ x 2

= lim Δ x \to 0 - sin (x + Δ x 2)

= - sin x

(lnx)′=1x

由导数的定义：

lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x

得：

(ln x)' = lim Δ x \to 0 ln ( x + Δ x ) - ln x Δ x

= lim Δ x \to 0 ln ( x + Δ x x ) Δ x

= lim Δ x \to 0 ln ( 1 + Δ x x ) Δ x x x

= 1 x

(logax)′=logaex

由导数的定义：

lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x

得：

(log a x)' = lim Δ x \to 0 log a ( x + Δ x ) - log a x Δ x

= lim Δ x \to 0 log a ( 1 + Δ x x ) Δ x

= lim Δ x \to 0 log a ( 1 + 1 x Δ x ) x Δ x Δ x x Δ x

= lim Δ x \to 0 log a e Δ x x Δ x

= lim Δ x \to 0 Δ x x log a e Δ x

= log a e x

(ax)′=axlna

由导数的定义：

lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x

得：

(a x)' = lim Δ x \to 0 a x + Δ x - a x Δ x

= lim Δ x \to 0 a x ( a Δ x - 1 ) Δ x

= a x lim Δ x \to 0 Δ x ln a Δ x

= a x ln a

(tanx)′=sec2x

(tan x)' = (sin x cos x)'

= ( sin x ) ' cos x - sin x ( cos x ) ' cos 2 x

= cos 2 x + sin 2 x cos 2 x

= sec 2 x

(cotx)′=−csc2x

(cot x)' = (1 tan x)'

= (cos x sin x)'

= ( cos x ) ' sin x - cos x ( sin x ) ' sin 2 x

= - csc 2 x

(secx)′=secxtanx

(sec x)' = (1 cos x)'

= 1 ' cos x - 1 ( cos x ) ' cos 2 x

= sin x cos 2 x

= sec x tan x

(cscx)′=−cscxcotx

(csc x)' = (1 sin x)'

= 1 ' sin x - 1 ( sin x ) ' sin 2 x

= - cos x sin 2 x

= - csc x cot x

(arcsinx)′=11−x2√

设 y = arcsin x ， 则 x = sin y

(arcsin x)' = (1 sin y)'

= 1 cos y

= 1 1 - sin 2 y - - - - - - - - \sqrt

= 1 1 - x 2 - - - - - \sqrt

(arccosx)′=−11−x2√

设 y = arccos x ， 则 x = cos y

(arccos x)' = (1 cos y)'

= - 1 sin y

= - 1 1 - cos 2 y - - - - - - - - \sqrt

= - 1 1 - x 2 - - - - - \sqrt

(arctanx)′=11+x2

设 y = arctan x ， 则 x = tan y

(arctan x)' = (1 tan y)'

= 1 sec 2 y

= 1 1 + tan 2 y

= 1 1 + x 2

(arccotx)′=−11+x2

设 y = a r c c o t x ， 则 x = cot y

(a r c c o t x)' = (1 cot y)'

= - 1 csc 2 y

= - 1 1 + cot 2 y

= - 1 1 + x 2

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宠物小精灵是什么软件

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