微分求解问题之乘法法则、商法则和链式求导法则
微分求解问题常用的三个基本法则是乘积法则、商法则和链式求导法则。下面是它们的公式和一些例子：
乘积法则
乘积法则用于求两个函数的乘积的导数。假设 u ( x ) u(x) u(x) 和 v ( x ) v(x) v(x) 是两个可微函数，则它们乘积的导数是：
( u ( x ) v ( x ) ) ′ = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x ) (u(x)v(x))' = u'(x)v(x) + u(x)v'(x) (u(x)v(x))′=u′(x)v(x)+u(x)v′(x)
示例
设 u ( x ) = x 2 u(x) = x^2 u(x)=x2， v ( x ) = e x v(x) = e^x v(x)=ex，则
( x 2 e x ) ′ = ( x 2 ) ′ e x + x 2 ( e x ) ′ = 2 x e x + x 2 e x = x e x ( 2 + x ) (x^2 e^x)' = (x^2)' e^x + x^2 (e^x)' = 2x e^x + x^2 e^x = x e^x (2 + x) (x2ex)′=(x2)′ex+x2(ex)′=2xex+x2ex=xex(2+x)
商法则
商法则用于求两个函数的商的导数。假设 u ( x ) u(x) u(x) 和 v ( x ) v(x) v(x) 是两个可微函数，且 v ( x ) ≠ 0 v(x) \neq 0 v(x)=0，则它们商的导数是：
( u ( x ) v ( x ) ) ′ = u ′ ( x ) v ( x ) − u ( x ) v ′ ( x ) v ( x ) 2 \left( \frac{u(x)}{v(x)} \right)' = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2} (v(x)u(x))′=v(x)2u′(x)v(x)−u(x)v′(x)
示例
设 u ( x ) = x 2 u(x) = x^2 u(x)=x2， v ( x ) = e x v(x) = e^x v(x)=ex，则
( x 2 e x ) ′ = ( x 2 ) ′ e x − x 2 ( e x ) ′ e 2 x = 2 x e x − x 2 e x e 2 x = e x ( 2 x − x 2 ) e 2 x = 2 x − x 2 e x \left( \frac{x^2}{e^x} \right)' = \frac{(x^2)' e^x - x^2 (e^x)'}{e^{2x}} = \frac{2x e^x - x^2 e^x}{e^{2x}} = \frac{e^x (2x - x^2)}{e^{2x}} = \frac{2x - x^2}{e^x} (exx2)′=e2x(x2)′ex−x2(ex)′=e2x2xex−x2ex=e2xex(2x−x2)=ex2x−x2
链式求导法则
链式求导法则用于求复合函数的导数。假设 y = f ( g ( x ) ) y = f(g(x)) y=f(g(x))，其中 f f f 和 g g g 都是可微函数，则
y ′ = f ′ ( g ( x ) ) ⋅ g ′ ( x ) y' = f'(g(x)) \cdot g'(x) y′=f′(g(x))⋅g′(x)
示例
设 f ( x ) = sin x f(x) = \sin x f(x)=sinx， g ( x ) = x 2 g(x) = x^2 g(x)=x2，则 y = sin ( x 2 ) y = \sin(x^2) y=sin(x2)，
y ′ = d d x sin ( x 2 ) = cos ( x 2 ) ⋅ d d x x 2 = cos ( x 2 ) ⋅ 2 x = 2 x cos ( x 2 ) y' = \frac{d}{dx} \sin(x^2) = \cos(x^2) \cdot \frac{d}{dx} x^2 = \cos(x^2) \cdot 2x = 2x \cos(x^2) y′=dxdsin(x2)=cos(x2)⋅dxdx2=cos(x2)⋅2x=2xcos(x2)
组合应用
设 h ( x ) = x 2 sin x e x h(x) = \frac{x^2 \sin x}{e^x} h(x)=exx2sinx，使用以上法则求导：
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先用商法则：
h ( x ) = u ( x ) v ( x ) , u ( x ) = x 2 sin x , v ( x ) = e x h(x) = \frac{u(x)}{v(x)}, \quad u(x) = x^2 \sin x, \quad v(x) = e^x h(x)=v(x)u(x),u(x)=x2sinx,v(x)=ex
h ′ ( x ) = u ′ ( x ) v ( x ) − u ( x ) v ′ ( x ) v ( x ) 2 h'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{v(x)^2} h′(x)=v(x)2u′(x)v(x)−u(x)v′(x) -
再用乘积法则求 u ( x ) u(x) u(x) 的导数：
u ′ ( x ) = ( x 2 sin x ) ′ = ( x 2 ) ′ sin x + x 2 ( sin x ) ′ = 2 x sin x + x 2 cos x u'(x) = (x^2 \sin x)' = (x^2)' \sin x + x^2 (\sin x)' = 2x \sin x + x^2 \cos x u′(x)=(x2sinx)′=(x2)′sinx+x2(sinx)′=2xsinx+x2cosx -
v ( x ) = e x v(x) = e^x v(x)=ex 的导数是：
v ′ ( x ) = e x v'(x) = e^x v′(x)=ex -
合并结果：
h ′ ( x ) = ( 2 x sin x + x 2 cos x ) e x − x 2 sin x ⋅ e x ( e x ) 2 = e x ( 2 x sin x + x 2 cos x − x 2 sin x ) e 2 x = 2 x sin x + x 2 cos x − x 2 sin x e x = x sin x + x 2 cos x e x h'(x) = \frac{(2x \sin x + x^2 \cos x)e^x - x^2 \sin x \cdot e^x}{(e^x)^2} = \frac{e^x (2x \sin x + x^2 \cos x - x^2 \sin x)}{e^{2x}} = \frac{2x \sin x + x^2 \cos x - x^2 \sin x}{e^x} = \frac{x \sin x + x^2 \cos x}{e^x} h′(x)=(ex)2(2xsinx+x2cosx)ex−x2sinx⋅ex=e2xex(2xsinx+x2cosx−x2sinx)=ex2xsinx+x2cosx−x2sinx=exxsinx+x2cosx
这样，我们就利用乘积法则、商法则和链式求导法则对复合函数进行了求导。