高数 02.02函数的求导法则

第二章第二节函数的求导法则

一、四则运算的求导法则

二、复合函数的求导法则

三、初等函数的求导问题

思路：

f ′ (x)=lim Δx→0 f(x+Δx)−f(x)Δx (构造性定义)

⇓ ⇓

求导法则⇓

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (C) ′ =0(sinx) ′ =cosx(lnx) ′ =1x ⎫ ⎭ ⎬ 证明中利用了两个重要极限 其他基本初等函数求导公式                                                        初等函数求导问题

一、四则运算的求导法则

定理1.函数u=u(x)及v=v(x)都在x具有导数⟹u(x)及v(x)的和、差、积、商(除分母为0的点外)都在点x可导,且(1)[u(x)±v(x)] ′ =u ′ (x)±v ′ (x)(2)[u(x)v(x)] ′ =u ′ (x)v(x)+u(x)v ′ (x)(3)[u(x)v(x) ] ′ =u ′ (x)v(x)−u(x)v ′ (x)v 2 x (v(x)≠0)用导数的定义可以证明上面的3个公式

推论：1)(Cu) ′ =Cu ′ (C为常数)2)(uvw) ′ =u ′ vw+uv ′ w+uvw ′ 3)(log a x) ′ =(lnxlna ) ′ =1xlna 4)(Cv ) ′ =−Cv ′ v 2 (C为常数)5)(u+v−w) ′ =u ′ +v ′ −w ′

例1.y=x √ (x 3 −4cosx−sin1),求y ′ 及y ′ ∣ ∣ x=1 .

解：y ′ =12 x 3 −4cosx−sin1x √ +x √ (3x 2 +4sinx−0)=7x 3 +8xsinx−4cosx−sin12x √ y ′ ∣ ∣ x=1 =7+7sin1−4cos12 =72 +72 sin1−2cos1

例2.求证(tanx) ′ =sec 2 x,(cscx) ′ =−cscxcotx.

证：(tanx) ′ =(sinxcosx ) ′ =sin ′ xcosx−sinxcos ′ xcos 2 x =cos 2 x+sin 2 xcos 2 x =1cos 2 x =sec 2 x (cscx) ′ =(1sinx ) ′ =−(sinx) ′ sin 2 x =−cosxsinxsinx =−cscxcotx 类似可证：(cotx) ′ =−csc 2 x,(secx) ′ =secxtanx

二、复合函数的求导法则

定理2.u=g(x)在点x可导,y=f(u)在点u=g(x)可导⟹复合函数y=f[g(x)]在点x可导，且dydx =f ′ (u)g ′ (x)

证：∵y=f(u)在点u可导，故lim Δu→0 ΔyΔx =f ′ (u)即ΔyΔu =f ′ (u)+α∴Δy=f ′ Δu+αΔu(当Δu→0时α→0)故有ΔyΔx =f ′ (u)ΔuΔx +αΔuΔx (Δx≠0)∴dydx =lim Δx→0 ΔyΔx =lim Δx→0 [f ′ (u)ΔuΔx +αΔuΔx ]=f ′ (u)g ′ (x)

推广：此法则可推广到多个中间变量的情形

例如：y=f(u),u=φ(v),v=ψ(x)dydx =dydu ⋅dudv ⋅dvdx =f ′ (u)⋅φ ′ (v)⋅ψ ′ (x)

例3.求下列导数：(1)(x μ ) ′ ;(2)(x x ) ′ ;(3)(sinhx) ′ .

解：(1)(x μ ) ′ =(e μlnx ) ′ =e μlnx ⋅(μlnx)′=x μ ⋅μx =μx μ−1 (2)(x x ) ′ =(e xlnx ) ′ =(e xlnx ) ′ ⋅(xlnx) ′ =x x (lnx+1)(3)(sinhx) ′ =(e x −e −x 2 ) ′ =e x +e −x 2 =coshx说明：类似可得(coshx) ′ =sinhx;(tanhx) ′ =1cosh 2 x ;(a x ) ′ =e xlna

例4.设y=lncos(e x ),求dydx .

解：dydx =1cos(e x ) ⋅(−sin(e x ))⋅e x =−e x tan(e x )

三、初等函数的求导问题

1.常数和基本初等函数的导数(P94)

(C) ′ =0(x μ ) ′ =μx μ−1 (sinx) ′ =cosx(cosx) ′ =−sinx(tanx) ′ =sec 2 x(cotx) ′ =−csc 2 x(secx) ′ =secxtanx(cscx) ′ =−cscxcotx(a x ) ′ =a x lna(e x ) ′ =e x (log a x) ′ =1xlna (lnx) ′ =1x

2.有限次四则运算的求导法则

(u±v) ′ =u ′ ±v ′ (Cu) ′ =Cu ′ (uv) ′ =u ′ v+uv ′ (uv ) ′ =u ′ v−uv ′ v 2 (v≠0)

3.复合函数的求导法则

y=f(u),u=φ(x)dydx =dydu ⋅dudx =f ′ (u)⋅φ ′ (x)

4.初等函数在定义域区间内可导，且导数仍为初等函数

说明：最基本的公式：

(C) ′ =0(sinx) ′ =cosx(lnx) ′ =1x

内容小结：

求导公式及求导法则(P94)

注意:

1)(uv) ′ ≠u ′ v ′ ,(uv ) ′ ≠u ′ v ′

2)搞清楚复合函数结构，有外向内逐层求导.