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课程表

导数的概念

函数的求导法则

隐函数及由参数方程确定的函数的导数

函数的微分

函数的求导法则

函数的和、差、积、商的求导法则

定理1 如果函数u=u(x)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)

(1)[u(x)±v(x)]

=limΔx0[u(x+Δx)±v(x+Δx)][u(x)±v(x)]Δx

=limΔx0u(x+Δx)±v(x+Δx)u(x)v(x)Δx

=limΔx0u(x+Δx)u(x)±v(x+Δx)v(x)Δx

=limΔx0[u(x+Δx)u(x)]±[v(x+Δx)v(x)]Δx

=limΔx0[u(x+Δx)u(x)Δx±v(x+Δx)v(x)Δx]

=limΔx0u(x+Δx)u(x)Δx±limΔx0v(x+Δx)v(x)Δx

=u(x)±v(x)

(2)[u(x)v(x)]

=limΔx0u(x+Δx)v(x+Δx)u(x)v(x)Δx

=limΔx0[u(x+Δx)v(x+Δx)Δxu(x)v(x)Δx+u(x)v(x+Δx)Δxu(x)v(x+Δx)Δx]

=limΔx0[u(x+Δx)u(x)Δxv(x+Δx)+u(x)v(x+Δx)v(x)Δx] =limΔx0[u(x+Δx)u(x)Δxv(x+Δx)]+limΔx0u(x)v(x+Δx)v(x)Δx = limΔx0u(x+Δx)u(x)ΔxlimΔx0v(x+Δx)+limΔx0u(x)limΔx0v(x+Δx)v(x)Δx

因为u(x)v(x)x点可导,因此它们在x点也连续,所以limΔx0v(x+Δx)=v(x),u(x)是常数,limΔx0u(x)=u(x).

v(x)=C时,[u(x)v(x)]=[Cu(x)]=Cu(x)

(3)[u(x)/v(x)]

=limΔx0u(x+Δx)v(x+Δx)u(x)v(x)Δx

=limΔx0u(x+Δx)v(x)v(x+Δx)u(x)v(x+Δx)v(x)Δx

=limΔx0[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)Δxu(x)v(x)v(x+Δx)v(x)Δx]

=limΔx0[u(x+Δx)v(x)v(x+Δx)v(x)Δxv(x+Δx)u(x)v(x+Δx)v(x)Δx+u(x)v(x)v(x+Δx)v(x)Δxu(x)v(x)v(x+Δx)v(x)Δx]

=limΔx0[u(x+Δx)u(x)v(x+Δx)v(x)Δxv(x)v(x+Δx)v(x)v(x+Δx)v(x)Δxu(x)] =limΔx0u(x+Δx)u(x)v(x+Δx)v(x)Δxv(x)limΔx0v(x+Δx)v(x)v(x+Δx)v(x)Δxu(x) =limΔx0u(x+Δx)u(x)Δxv(x)v(x+Δx)v(x)limΔx0v(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)