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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)u = u(x) 及 v=v(x)v = v(x) 都在点 xx 具有导数，那么它们的和、差、积、商（除分母为零的点外）都在 xx 具有导数，且：

  1. [u(x)±v(x)]′=u′(x)±v′(x)\LARGE [u(x) \pm v(x)]' = u'(x) \pm v'(x)
  2. [u(x)v(x)]′=u′(x)v(x)+u(x)v′(x)\LARGE [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)\LARGE [\frac{u(x)}{v(x)}]' = \frac{u'(x) v(x) - u(x) v'(x)}{v^2(x)} (v(x) \not =0)

反函数的求导法则​

定理 2​

如果函数 x=f(y)x = f(y) 在区间 IyI_y 内单调、可导且 f′(y)≠0f'(y) \not = 0，那么它的反函数 y=f−1(x)y = f^{-1}(x) 在区间 Ix=x∣x=f(y),y∈IyI_x = {x | x = f(y), y \in I_y} 内也可导，且：

[f−1(x)]′=1f′(y)或dydx=1dxdy\LARGE [f^{-1}(x)]' = \frac{1}{f'(y)} \text{或} \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}

复合函数求导法则​

定理 3​

如果 u=g(x)u = g(x) 在点 xx 可导，而 y=f(u)y = f(u) 在点 u=g(x)u = g(x) 可导，那么复合函数 y=f[g(x)]y = f[g(x)] 在点 xx 可导，且其导数为：

dydx=f′(u)⋅g′(x)或dydx=dydu⋅dudx\LARGE \frac{dy}{dx} = f'(u) · g'(x) \text{或} \frac{dy}{dx} = \frac{dy}{du} · \frac{du}{dx}

基本求导法则与导数公式​

常数和基本初等函数的导数公式​

  1. (C)′=0\LARGE (C)' = 0
  2. (xμ)′=μxμ−1\LARGE (x^\mu)' = \mu x^{\mu - 1}
  3. (sin⁡x)′=cos⁡x\LARGE (\sin x)' = \cos x
  4. (cos⁡x)′=−sin⁡x\LARGE (\cos x)' = -\sin x
  5. (tan⁡x)′=sec⁡2x\LARGE (\tan x)' = \sec^2 x
  6. (cot⁡x)′=−csc⁡2x\LARGE (\cot x)' = -\csc^2 x
  7. (sec⁡x)′=sec⁡xtan⁡x\LARGE (\sec x)' = \sec x\tan x
  8. (csc⁡x)′=−csc⁡xcot⁡x\LARGE (\csc x)' = -\csc x\cot x
  9. (ax)′=axln⁡a(a>0,a≠1)\LARGE (a^x)' = a^x \ln a (a > 0, a \not = 1)
  10. (ex)′=ex\LARGE (e^x)' = e^x
  11. (log⁡ax)′=1xln⁡a(a>0,a≠1)\LARGE (\log_a x)' = \frac{1}{x \ln a} (a > 0, a \not = 1)
  12. (ln⁡x)′=1x\LARGE (\ln x)' = \frac{1}{x}
  13. (arcsin⁡x)′=11−x2\LARGE (\arcsin x)' = \frac{1}{\sqrt{1 - x^2}}
  14. (arccos⁡x)′=−11−x2\LARGE (\arccos x)' = -\frac{1}{\sqrt{1 - x^2}}
  15. (arctan⁡x)′=11+x2\LARGE (\arctan x)' = \frac{1}{1 + x^2}
  16. (arccotx)′=−11+x2\LARGE (\text{arccot} x)' = -\frac{1}{1 + x^2}

函数的和、差、积、商的求导法则​

设 u=u(x),v=(x)u = u(x), v = (x) 都可导，则：

  1. (u±v)′=u′±v′\LARGE (u \pm v)' = u' \pm v'
  2. (Cu)′=Cu′\LARGE (Cu)' = Cu' （CC 是常数）
  3. (uv)′=u′v+uv′\LARGE (uv)' = u'v + uv'
  4. (uv)′=u′v−uv′v2\LARGE (\frac{u}{v})' = \frac{u'v -uv'}{v^2} （v≠0v \not = 0）