江画工太猩院 第2节 函数的求导法则
江西理工大学理学院 第 2 节 函数的求导法则
江画工太猩院 和、差、积、商的求导法则 定理如果函数m(x),v(x)在点处可导,则它 们的和、差、积、商(分母不为零在点x处也 可导,并且 (1)[(x)±v(x)=u(x)土v(x) (2)|u(x)v(x)"=l'(x)y(x)+(x)y(x) (3) u(x) u(xv(x-u(xv(r) (v(x)≠0)
江西理工大学理学院 一、和、差、积、商的求导法则 定理 可导 并且 们的和、差、积、商 分母不为零 在点 处也 如果函数 在点 处可导 则它 , ( ) ( ), ( ) , x u x v x x ( ( ) 0). ( ) ( ) ( ) ( ) ( ) ] ( ) ( ) ( 3 ) [ ( 2 ) [ ( ) ( ) ] ( ) ( ) ( ) ( ); ( 1 ) [ ( ) ( ) ] ( ) ( ); 2 ≠ ′ − ′ ′ = ⋅ ′ = ′ + ′ ± ′ = ′ ± ′ v x v x u x v x u x v x v x u x u x v x u x v x u x v x u x v x u x v x
江画猩工式塑辱院 证(1)略. 证(2)设∫(x)=u(x)(x) cla f(x+h-f(r) x=lim- h→0 lim (x+h)v(x+ h)-u().v(r) b-→0 im u(x+h)(x+h)=u(x) v(x+h) h h-0 +u(x) v(x+h)-u(x)(xl
江西理工大学理学院 证(2) 设 f (x) = u(x)v(x), h f x h f x f x h ( ) ( ) ( ) lim0 + − ′ = → ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( ) 1 lim 0 u x v x h u x v x u x h v x h u x v x h h h + ⋅ + − ⋅ = + ⋅ + − ⋅ + → h u x h v x h u x v x h ( ) ( ) ( ) ( ) lim 0 + ⋅ + − ⋅ = → 证(1) 略
江画工太猩院 limr u(r+h)-u(x v(x+h) h vlx+h x +(x). lim u(rt h -u(x limb(x+ h h-0 h-0 +以(x)lim v(x+h)-v(r h->0 h u(x)·v(x)+(x)v(x)
江西理工大学理学院 ] ( ) ( ) ( ) ( ) ( ) ( ) lim[ 0 h v x h v x u x v x h h u x h u x h + − + ⋅ ⋅ + + − = → = u ′(x)⋅ v(x) + u(x)⋅ v′(x) h v x h v x u x v x h h u x h u x h h h ( ) ( ) ( ) lim lim ( ) ( ) ( ) lim 0 0 0 + − + ⋅ ⋅ + + − = → → →
江画工太猩院 证(3)设(x)=“x,(x)≠0 ∫(x)=lim .∫(x+h)-f(x) h→0h u(x+h u(x) lim v(x+h)v(r) h→0 =lim m-th)v(r)-u(x]v(+h) h→>0 v(x+hv(r)h
江西理工大学理学院 证(3) , ( ( ) 0), ( ) ( ) ( ) = v x ≠ v x u x 设 f x h f x h f x f x h ( ) ( ) ( ) lim0 + − ′ = → v x h v x h u x h v x u x v x h h ( ) ( ) ( ) ( ) ( ) ( ) lim 0 + + − + = → h v x u x v x h u x h h ( ) ( ) ( ) ( ) lim 0 − + + = →