函数的和、差、积、商的求导法则
定理1 如果函数u=u(x)及v=v(x)都在点x具有导数,那么它们的和、差、积、商(除分母为零的点外)都在点x具有导数,且函数的和、差、积、商的求导法则
(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)v2(x)(v(x)≠0)
证
(1)[u(x)±v(x)]′
=limΔx→0[u(x+Δx)±v(x+Δx)]−[u(x)±v(x)]Δx
=limΔx→0u(x+Δx)±v(x+Δx)−u(x)∓v(x)Δx
=limΔx→0u(x+Δx)−u(x)±v(x+Δx)∓v(x)Δx
=limΔx→0[u(x+Δx)−u(x)]±[v(x+Δx)−v(x)]Δx
=limΔx→0[u(x+Δx)−u(x)Δx±v(x+Δx)−v(x)Δx]
=limΔx→0u(x+Δx)−u(x)Δx±limΔx→0v(x+Δx)−v(x)Δx
=u′(x)±v′(x)
(2)[u(x)v(x)]′
=limΔx→0u(x+Δx)v(x+Δx)−u(x)v(x)Δx=limΔx→0[u(x+Δx)v(x+Δx)Δx−u(x)v(x)Δx+u(x)v(x+Δx)Δx−u(x)v(x+Δx)Δx]
=limΔx→0[u(x+Δx)−u(x)Δxv(x+Δx)+u(x)v(x+Δx)−v(x)Δx] =limΔx→0[u(x+Δx)−u(x)Δxv(x+Δx)]+limΔx→0u(x)v(x+Δx)−v(x)Δx = limΔx→0u(x+Δx)−u(x)ΔxlimΔx→0v(x+Δx)+limΔx→0u(x)limΔx→0v(x+Δx)−v(x)Δx因为u(x)和v(x)在x点可导,因此它们在x点也连续,所以limΔx→0v(x+Δx)=v(x),u(x)是常数,limΔx→0u(x)=u(x).
当v(x)=C时,[u(x)v(x)]′=[Cu(x)]′=Cu′(x)
(3)[u(x)/v(x)]′
=limΔx→0u(x+Δx)v(x+Δx)−u(x)v(x)Δx =limΔx→0u(x+Δx)v(x)−v(x+Δx)u(x)v(x+Δx)v(x)Δx=limΔx→0[u(x+Δx)v(x)−v(x+Δx)u(x)v(x+Δx)v(x)Δx+u(x)v(x)v(x+Δx)v(x)Δx−u(x)v(x)v(x+Δx)v(x)Δx]
=limΔx→0[u(x+Δx)v(x)v(x+Δx)v(x)Δx−v(x+Δx)u(x)v(x+Δx)v(x)Δx+u(x)v(x)v(x+Δx)v(x)Δx−u(x)v(x)v(x+Δx)v(x)Δx]
=limΔx→0[u(x+Δx)−u(x)v(x+Δx)v(x)Δxv(x)−v(x+Δx)−v(x)v(x+Δx)v(x)Δxu(x)] =limΔx→0u(x+Δx)−u(x)v(x+Δx)v(x)Δxv(x)−limΔx→0v(x+Δx)−v(x)v(x+Δx)v(x)Δxu(x) =limΔx→0u(x+Δx)−u(x)Δxv(x)v(x+Δx)v(x)−limΔx→0v(x+Δx)−v(x)Δxu(x)v(x+Δx)v(x) =u′(x)v(x)v2(x)−v′(x)u(x)v2(x) =u′(x)v(x)−v′(x)u(x)v2(x)