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导数的加法与减法法则_图文

求函数的导数的步骤是怎样的?
(1)求函数的增量?y ? f (x ? ?x) ? f (x);

(2)求函数的增量与自变量的增量的比值 :

?y ? f (x ? ?x) ? f (x) ;

?x

?x

(3)求极限,得导函数y? ? f ?(x) ? lim ?y . ?x?0 ?x

导数公式表(其中三角函数的自变量单位是弧度)

如果已知两个函数的导数,如何求这两个 函数的和,差,积,商的导数呢?
给出函数 f (x) ? x ? x2,如何来求这个函数的导 函数 ?

实例分析

按照求函数导数的步骤: 首先给定自变量x一个改变量?x, 则函数值y的改变量为 ?y ? f (x ? ?x) ? f (x) ? (x ? ?x) ? (x ? ?x)2 ? (x ? x2 ) ? ?x ? 2x?x ? ?x2.

相应的平均变化率为

?y ? ?x ? 2x?x ? ?x2 ? 1? 2x ? ?x.

?x

?x

当?x趋于0时,得到导数 : f ?(x) ? 1? 2x.

可以看出 : (x ? x2 )? ? x? ? (x2 )?.

两个函数和(差)的导数等于这两个函数 导数的和(差),即:
? f (x) ? g(x)?? ? f ?(x) ? g?(x), ? f (x) ? g(x)?? ? f ?(x) ? g?(x).

例1求下列函数的导数: (1) y ? x2 ? 2;(2) y ? x ? ln x.
解 : (1)函数y ? x2 ? 2x是f (x) ? 2x与 函数g(x) ? 2x的和,由导数公式表, 分别得出 f ?(x) ? 2x, g?(x) ? 2x ln 2. 利用函数和的求导法则可得 :
? ? x2 ? 2x ? ? f ?(x) ? g?(x) ? 2x ? 2x ln 2.

(2)函数y ? x ? ln x是f (x) ? x与

函数g(x) ? ln x的差,由导数公式表,

分别得出

f ?(x) ? 1 , g?(x) ? 1 .

2x

x

利用函数差的求导法则可得 :

? ?? x ? ln x

?

f ?(x) ? g?(x) ?

1

? 1.

2x x

提示:
对于常用的几 个函数的导数, 可以熟记,以便 以后使用.

?例2求曲线y ? x3 ? 1 过点的切线方程.

x

解: 首先求出函数y ? x3 ? 1 在x ? 1处的导数.

函数y

?

x3

?

1

是函数f

(x)

x ?

x 3与g ( x)

?

1

的差,

x

x

由导数公式表分别得出

f

?(

x)

?

3x

2

,

g?(

x)

?

?

1 x2

.

根据函数差的求导法则可得 :

?? x3 ?

?

1 ??? x?

?

f ?(x) ? g?(x)

? 3x2

?

1 x2

.

将x ? 1代入导函数得
3?1? 1 ? 4 1
即曲线y ? x3 ? 1 过点(1,0)的切线斜率为4, x
从而其切线方程为
y ? 0 ? 4(x ?1) 即: y ? 4(x ?1).

练习:如图已知曲线

y

?

1 3

x 3上 一 点P (2,

8) 3

,求:

(1)点P处的切线的斜率; (2)点P处的切线方程.

解 : (1)由导数公式得: f ?(x) ? ?? 1 x3 ??? ? 1 ?3? x2 ? x2. ?3 ? 3
故点P的切线斜率是: f ?(x0 ) ? 22 ? 4.
(2)点P处的切线方程为: y ? 2 ? 4?? x ? 8 ?? ? 3?
即: 3y ?12x ? 26 ? 0.

导数的加法与减法法则是什么?
几个常用的函数的导数是什么?
y ? c(c是常数), y ? x? (?为实数),
y ? ax (a ? 0, a ? 1), y ? log a x(a ? 0, a ? 1), y ? sin x, y ? cosx, y ? tan x, y ? cot x.


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