和差积商 · 复合
基本求导公式与运算法则
逐个用定义式算导数很快就不划算：f ( x ) = x 5 sin x f(x) = x^5 \sin x f ( x ) = x 5 sin x x 5 x^5 x 5 sin x \sin x sin x
1 · 基本初等函数的导数
函数
导函数
备注
f ( x ) = c f(x) = c f ( x ) = c
f ′ ( x ) = 0 f'(x) = 0 f ′ ( x ) = 0
常数函数的图象是水平直线
f ( x ) = x α f(x) = x^\alpha f ( x ) = x α
f ′ ( x ) = α x α − 1 f'(x) = \alpha x^{\alpha - 1} f ′ ( x ) = α x α − 1
α \alpha α x x x
f ( x ) = sin x f(x) = \sin x f ( x ) = sin x
f ′ ( x ) = cos x f'(x) = \cos x f ′ ( x ) = cos x
弧度制下成立
f ( x ) = cos x f(x) = \cos x f ( x ) = cos x
f ′ ( x ) = − sin x f'(x) = -\sin x f ′ ( x ) = − sin x
多出的负号来自余弦的减少
f ( x ) = a x f(x) = a^x f ( x ) = a x
f ′ ( x ) = a x ln a f'(x) = a^x \ln a f ′ ( x ) = a x ln a
a > 0 a > 0 a > 0 a ≠ 1 a \ne 1 a = 1
f ( x ) = e x f(x) = e^x f ( x ) = e x
f ′ ( x ) = e x f'(x) = e^x f ′ ( x ) = e x
ln e = 1 \ln e = 1 ln e = 1
f ( x ) = log a x f(x) = \log_a x f ( x ) = log a x
f ′ ( x ) = 1 x ln a f'(x) = \dfrac{1}{x \ln a} f ′ ( x ) = x ln a 1
x > 0 x > 0 x > 0
f ( x ) = ln x f(x) = \ln x f ( x ) = ln x
f ′ ( x ) = 1 x f'(x) = \dfrac{1}{x} f ′ ( x ) = x 1
同上取
a = e a = e a = e
幂函数一栏把整数、分数、负数指数统一了：x = x 1 / 2 \sqrt{x} = x^{1/2} x
= x 1/2 1 2 x − 1 / 2 = 1 2 x \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}} 2 1 x − 1/2 = 2 x
1 1 x = x − 1 \frac{1}{x} = x^{-1} x 1 = x − 1 − x − 2 -x^{-2} − x − 2 x → 0 + x \to 0^+ x → 0 + x \sqrt{x} x
1 x \frac{1}{x} x 1
e e e a x a^x a x ln a \ln a ln a a = e a = e a = e 1 1 1 e x e^x e x
2 · 四则运算的求导法则
定理 2.1 设
u = u ( x ) u = u(x) u = u ( x ) v = v ( x ) v = v(x) v = v ( x )
( u ± v ) ′ = u ′ ± v ′ , ( u v ) ′ = u ′ v + u v ′ , ( u v ) ′ = u ′ v − u v ′ v 2 ( v ≠ 0 ) (u \pm v)' = u' \pm v', \qquad (uv)' = u'v + uv', \qquad \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \;\; (v \ne 0) ( u ± v ) ′ = u ′ ± v ′ , ( uv ) ′ = u ′ v + u v ′ , ( v u ) ′ = v 2 u ′ v − u v ′ ( v = 0 )
和差一栏形式上最简单，积与商都不是。( u v ) ′ = u ′ v + u v ′ (uv)' = u'v + uv' ( uv ) ′ = u ′ v + u v ′ ( u v ) ′ = u ′ v ′ (uv)' = u'v' ( uv ) ′ = u ′ v ′ u = v = x u = v = x u = v = x ( x 2 ) ′ = 2 x (x^2)' = 2x ( x 2 ) ′ = 2 x 1 ⋅ 1 = 1 1 \cdot 1 = 1 1 ⋅ 1 = 1 u ′ v − u v ′ u'v - uv' u ′ v − u v ′ u v ′ − u ′ v uv' - u'v u v ′ − u ′ v
常数倍是积法则的特例：( c u ) ′ = c ′ u + c u ′ = c u ′ (cu)' = c'u + cu' = cu' ( c u ) ′ = c ′ u + c u ′ = c u ′ c ′ = 0 c' = 0 c ′ = 0 ( 3 x 4 − 5 x + 2 ) ′ = 12 x 3 − 5 (3x^4 - 5x + 2)' = 12x^3 - 5 ( 3 x 4 − 5 x + 2 ) ′ = 12 x 3 − 5
图 2-1 · 两个基本初等函数按四条法则合成，右侧给出法则算出的导函数与中心差商的逐点对照。可切换
u u u v v v x x x u ′ v ′ u'v' u ′ v ′
3 · 复合函数与链式法则
y = sin ( 2 x + 1 ) y = \sin(2x + 1) y = sin ( 2 x + 1 ) y = sin u y = \sin u y = sin u u = 2 x + 1 u = 2x + 1 u = 2 x + 1
定理 3.1（链式法则） 设
y = f ( u ) y = f(u) y = f ( u ) u = g ( x ) u = g(x) u = g ( x ) y = f ( g ( x ) ) y = f(g(x)) y = f ( g ( x )) y ′ = f ′ ( g ( x ) ) ⋅ g ′ ( x ) y' = f'(g(x)) \cdot g'(x) y ′ = f ′ ( g ( x )) ⋅ g ′ ( x ) d y d x = d y d u ⋅ d u d x \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} d x d y = d u d y ⋅ d x d u
莱布尼茨记号在链式法则上最见长：形式上像两个分数相乘，d u du d u y = sin ( 2 x + 1 ) y = \sin(2x + 1) y = sin ( 2 x + 1 ) cos ( 2 x + 1 ) ⋅ 2 \cos(2x+1) \cdot 2 cos ( 2 x + 1 ) ⋅ 2 2 2 2 y = e x 2 y = e^{x^2} y = e x 2 2 x e x 2 2x e^{x^2} 2 x e x 2 y = ln ( 3 x − 1 ) y = \ln(3x - 1) y = ln ( 3 x − 1 ) 3 3 x − 1 \frac{3}{3x - 1} 3 x − 1 3
例 3.1 求
y = sin x x y = \frac{\sin x}{x} y = x s i n x y = x 2 + 1 y = \sqrt{x^2 + 1} y = x 2 + 1
y ′ = cos x ⋅ x − sin x ⋅ 1 x 2 = x cos x − sin x x 2 y' = \frac{\cos x \cdot x - \sin x \cdot 1}{x^2} = \frac{x\cos x - \sin x}{x^2} y ′ = x 2 c o s x ⋅ x − s i n x ⋅ 1 = x 2 x c o s x − s i n x u \sqrt{u} u
u = x 2 + 1 u = x^2 + 1 u = x 2 + 1 y ′ = 1 2 x 2 + 1 ⋅ 2 x = x x 2 + 1 y' = \frac{1}{2\sqrt{x^2+1}} \cdot 2x = \frac{x}{\sqrt{x^2+1}} y ′ = 2 x 2 + 1
1 ⋅ 2 x = x 2 + 1
x
4 · 用差商核对法则
本系列的单测不把求导结果写成常量，而是让 combine 按法则现算——积走
u ′ v + u v ′ u'v + uv' u ′ v + u v ′ u ′ ( v ) ⋅ v ′ u'(v) \cdot v' u ′ ( v ) ⋅ v ′ u ′ v ′ u'v' u ′ v ′
核对里有两处要留一手。其一，商在分母接近零处差商本身就不可靠，sin x ln x \frac{\sin x}{\ln x} l n x s i n x x x x 1 1 1 x x x ln ( x 2 ) \ln(x^2) ln ( x 2 ) x = − 2 x = -2 x = − 2 ( 0 , + ∞ ) (0, +\infty) ( 0 , + ∞ ) x x x
注 · 把合成式回显成可读的表达式时，最初的写法是对外层解析式做字符串替换，把字面的 x 换成内层表达式。这个做法在
e x e^x e x x \sqrt{x} x
x 可替换，替换结果是原样的
e x e^x e x sub(inner) 模板，e^(x²) 这类回显由模板拼出，正确性也进了单测。
5 · 参考文献
Differentiation rules . Wikipedia. 基本导数表与和差积商法则的完整清单。https://en.wikipedia.org/wiki/Differentiation_rules Product rule . Wikipedia. 积法则的证明与常见误用。https://en.wikipedia.org/wiki/Product_rule Quotient rule . Wikipedia. 商法则的推导，以及由积法则加链式法则重新得到它的路径。https://en.wikipedia.org/wiki/Quotient_rule Chain rule . Wikipedia. 链式法则的表述、莱布尼茨记号的写法与多层复合。https://en.wikipedia.org/wiki/Chain_rule
上一篇 导数的几何意义与切线方程 下一篇 导数的符号与函数的单调性