目录
1、常数和基本初等函数的导数公式
公式 | 公式 |
---|---|
(1) \((C)' = 0\) | (2)\((x^{\mu})' = \mu x^{\mu - 1}\) |
(3)\((\sin x)' = \cos x\) | (4)\((\cos x)' = - \sin x\) |
(5)\((\tan x)' = \sec^2 x\) | (6)\((\cot x)' = - \csc^2 x\) |
(7)\((\sec x)' = \sec x \tan x\) | (8)\((\csc x)' = - \csc x \cot x\) |
(9)\((a^x)' = a^x \ln a\) | (10)\((\mathrm{e}^x)' = \mathrm{e}^x\) |
(11)\((\log_a x)' = \cfrac{1}{x \ln a}\) | (12)\((\ln x)’ = \cfrac{1}{x}\) |
(13)\((\arcsin x)' = \cfrac{1}{\sqrt{1 - x^2}}\) | (14)$$(\arccos x)' = - \cfrac{1}{\sqrt{1 - x^2}}$$ |
(15)\((\arctan x)’ = \cfrac{1}{1 + x^2}\) | (16)\((\operatorname{arccot} x)’ = - \cfrac{1}{1 + x^2}\) |
2、函数的和、差、积、商的求导法则
设 \(u = u(x), v = v(x)\) 都可导,则
(1)\((u \pm v)' = u' \pm v'\)
(2)\((C u)' = C u' (C是常数)\)
(3)\((u v)' = u' v + u v'\)
(4)\(\left( \cfrac{u}{v} \right)' = \cfrac{u' v - u v'}{v^2} (v \neq 0)\)
3、反函数的求导法则
设 \(x = f(y)\) 在区间 \(I_y\) 内单调、可导且 \(f' (y) \neq 0\) ,则它的反函数 \(y = f^{-1} (x)\) 在 \(I_x = f(I_y)\) 内也可导,且
\[[f^{-1}(x)]' = \cfrac{1}{f'(y)} \quad 或 \quad \cfrac{\mathrm d y}{\mathrm d x} = \cfrac{1}{\frac{\mathrm d x}{\mathrm d y}} . \]4、复合函数的求导法则
设 \(y = f(u)\) ,而 \(u = \mathrm g (x)\) 且 \(f(u)\) 及 \(\mathrm g (x)\) 都可导,则复合函数 \(y = f[\mathrm g (x)]\) 的导数为
\[\cfrac{\mathrm{d}y}{\mathrm{d}x} = \cfrac{\mathrm{d}y}{\mathrm{d}u} \cdot \cfrac{\mathrm{d}u}{\mathrm{d}x} \quad 或 \quad y'(x) = f'(u) \mathrm{g} '(x) . \]例 设 \(y = \sin{nx} \cdot \sin^n x\) (\(n\) 为常数),求 \(y'\) .
解首先应用积的求导法则得
在计算 \((\sin{nx})'\) 与 \((\sin^n x)'\) 时,都要应用复合函数的求导法则,由此得
\[\begin{align*} y' &= n \cos{nx} \cdot \sin^n x + \sin{nx} \cdot n \sin^{n - 1} x \cdot \cos x \\ &= n \sin^{n - 1} x (\cos{nx} \cdot \sin x + \sin{nx} \cdot \cos x) \\ &= n \sin^{n - 1} x \cdot \sin{(n + 1) x} \end{align*} \] 标签:cdot,nx,cfrac,求导,2.2,高等数学,sin,mathrm From: https://www.cnblogs.com/mowenpan1995/p/18414202/gdsx2-2hsdqdfz