第二节
导数的四则运算 基本初等函数的导数 复合函数的导数 反函数求导法则
导数的四则运算
（1）设 u ( x) v( x) 在x可导，则[u ( x) ± v( x)]′ = u ′( x) ± v′( x) 设 y = g ( x) = u ( x) + v( x)
Δy = g ( x + Δx) − g ( x) = [u ( x + Δx) + v( x + Δx)] − [u ( x) + v( x)]
= [u ( x + Δx) − u ( x)] + [v( x + Δx) − v( x)]
= Δu + Δv Δy Δu Δv lim Δy = lim [ Δu + Δv ] = u ′( x) + v′( x) + = Δx Δx Δx Δx →0 Δx Δx →0 Δx Δx
推广[u1 ( x) ± u2 ( x) ±
′ ( x) ± u2 ′ ( x) ± u n ( x)]′ = u1
′ ( x). ± un
[u ( x) ± v( x)]′ = u ′( x) ± v′( x)
[u1 ( x) ± u2 ( x) ±
例 解
指数函数求导′ ( x) ± u2 ′ ( x) ± u n ( x)]′ = u1
′ ( x). ± un
f ( x) = x + sin x − cos x + 9 求其导数 f ′( x) = ( x + sin x − cos x + 9)′ = ( x )′ + (sin x)′ − (cos x)′ + (9)′
= 1 / 2 x + cos x + sin x
(2)设u ( x) , v( x)在x可导，则[u ( x)v( x)]′ = u ( x)v′( x) + u ′( x)v ( x ) 设 y = g ( x ) = u ( x )v ( x )
Δy = g ( x + Δx ) − g ( x ) = u ( x + Δ x ) v ( x + Δx ) − u ( x ) v ( x ) = u ( x + Δ x ) v ( x + Δ x ) − u ( x ) v ( x + Δx ) + u ( x ) v ( x + Δ x ) − u ( x ) v ( x )
= Δu ⋅ v( x + Δx) + u ( x)Δv. Δv Δy Δu = v ( x + Δx ) + u ( x ) . Δx Δx Δx Δy Δu Δv lim = lim ⋅ lim v( x + Δx) + u ( x) ⋅ lim Δx → 0 Δx Δx → 0 Δx Δx → 0 Δx → 0 Δx
= u ( x)v′( x) + u ′( x)v( x).
[u ( x)v( x)]′ = u ( x)v′( x) + u ′( x)v ( x )
[cu ( x)]′ = cu ′( x) （常数因子可以提出来） 特别：
例、求 f (x) = 7 x cosx 的导数 解 f ′( x) = (7 x cos x)′ = 7[
( x ) cos x +
′
′ x (cos x ) ]
cos x = 7[ − x sin x] 2 x
推广 (u ( x)v( x) w( x))′
轮流求导
= u ′( x)v( x) w( x) + u ( x)v′( x) w( x) + u ( x)v( x) w′( x)
[u1 ( x)u2 ( x)
′ ( x)u2 ( x) un ( x)]′ = u1 ′ ( x) + u1 ( x)u 2 un ( x) +
un ( x) + u1 ( x)u2 ( x) ′ ( x). un
例、求 f ( x ) = 4 x 2 ⋅ ln x ⋅ cos x 的导数 解 f ′(x) = (4x2 ⋅ ln x ⋅ cosx)′ = 4(x2 ⋅ ln x ⋅ cosx)′
1 = 4(2x ⋅ ln x ⋅ cosx + x2 ⋅ ⋅ cosx − x2 ⋅ ln x ⋅ sin x) x = 4(2x ⋅ ln x ⋅ cosx + x cosx − x2 ⋅ ln x ⋅ sin x)