\frac{d}{dx} (x^n) = n x^{n-1}

\frac{d}{dx}x^n=\lim_{h\to 0}\frac{(x+h)^n-x^n}{h} \\[10pt] (x+h)^n = x^n + n x^{n-1}h + \frac{n(n-1)}{2}x^{n-2}h^2+\cdots \frac{(x+h)^n-x^n}{h}=n x^{n-1}+\frac{n(n-1)}{2}x^{n-2}h+\cdots \frac{d}{dx}x^n = n x^{n-1}

\frac{d}{dx} e^x = e^x

\frac{d}{dx}e^x =\lim_{h\to 0} \frac{e^{x+h} - e^x}{h} = e^x\lim_{h\to 0} \frac{e^h - 1}{h} \\[6pt] 而数学中有\quad \lim_{h\to 0} \frac{e^h - 1}{h}=1 \quad 于是\frac{d}{dx}e^x= e^x

\frac{d}{dx} \ln x = \frac{1}{x}

y=\ln x \Rightarrow x=e^y \\[6pt] 1=\frac{dy}{dx} e^y = \frac{dy}{dx} x \\[6pt] \frac{dy}{dx}=\frac{1}{x}

\frac{d}{dx}\sin x = \cos x

\frac{\sin(x+h)-\sin x}{h} = \sin x\frac{\cos h-1}{h}+\cos x\frac{\sin h}{h} \\[6pt] \lim_{h\to 0}\frac{\sin h}{h}=1,\quad \lim_{h\to 0}\frac{\cos h -1}{h}=0 \\[6pt] \frac{d}{dx}\sin x = \cos x

\frac{d}{dx}\cos x = -\sin x

\frac{\cos(x+h)-\cos x}{h} = -\sin x\frac{\sin h}{h} + \cos x\frac{\cos h -1}{h} \frac{d}{dx}\cos x = -\sin x

\frac{d}{dx} C = 0

\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

\frac{d}{dx}[uv] \\[6pt] = \lim_{h\to 0} \frac{u(x+h)v(x+h)-u(x)v(x)}{h} \\[6pt] = \lim_{h\to 0} \frac{u(x+h)v(x+h)-u(x)v(x+h)+u(x)v(x+h)-u(x)v(x)}{h} \\[6pt] = \lim_{h\to 0} \left[ v(x+h)\frac{u(x+h)-u(x)}{h} \\[6pt] + u(x)\frac{v(x+h)-v(x)}{h} \right] \\[6pt] = u'(x)v(x) + u(x)v'(x)

\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x)-u(x)v'(x)}{[v(x)]^2}

\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{d}{dx}[u \cdot v^{-1}] = u'(x)v^{-1} + u(x)(-1)v^{-2}v' = \frac{u'v - uv'}{v^2}

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

\frac{d}{dx}f(g(x)) \\[6pt] = \lim_{h\to 0} \frac{f(g(x+h)) - f(g(x))}{h} \\[6pt] 设 \Delta g = g(x+h) - g(x) \\[6pt] = \lim_{h\to 0} \frac{f(g(x)+\Delta g)-f(g(x))}{\Delta g} \cdot \frac{\Delta g}{h} \\[6pt] = f'(g(x)) \cdot g'(x)

\frac{d}{dx} (x^n) = n x^{n-1}\quad \frac{d}{dx} e^x = e^x\quad \frac{d}{dx} \ln x = \frac{1}{x}\quad \frac{d}{dx} \sin x = \cos x\quad \frac{d}{dx} \cos x = -\sin x\quad \frac{d}{dx} \tan x = \sec^2 x \\[6pt] \frac{d}{dx} \cot x = -\csc^2 x\quad \frac{d}{dx} \sec x = \sec x \tan x\quad \frac{d}{dx} \csc x = -\csc x \cot x\quad \frac{d}{dx} C = 0

\frac{d}{dx} \ln(ax+b) = \frac{a}{ax+b} \quad \frac{d}{dx} e^{kx} = k e^{kx} \quad \frac{d}{dx} \sin(kx) = k \cos(kx) \quad \frac{d}{dx} \cos(kx) = -k \sin(kx) \\[6pt] \frac{d}{dx} (a x^n + b x + c) = a n x^{n-1} + b

\frac{d}{dx}(x^n) = n x^{n-1} \quad \frac{d}{dx}(a^x) = a^x \ln a \quad \frac{d}{dx}(e^x) = e^x \quad \frac{d}{dx}(\ln x) = \frac{1}{x} \quad \frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}

\frac{d}{dx}(\sin x) = \cos x \quad \frac{d}{dx}(\cos x) = -\sin x \quad \frac{d}{dx}(\tan x) = \sec^2 x \quad \frac{d}{dx}(\cot x) = -\csc^2 x \quad \frac{d}{dx}(\sec x) = \sec x \tan x \quad \frac{d}{dx}(\csc x) = -\csc x \cot x \quad

\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}} \quad \frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 - x^2}} \quad \frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2} \quad \frac{d}{dx}(arccot x) = -\frac{1}{1 + x^2}

\frac{d}{dx}(\sinh x) = \cosh x \quad \frac{d}{dx}(\cosh x) = \sinh x \quad \frac{d}{dx}(\tanh x) = \text{sech}^2 x \quad \frac{d}{dx}(\coth x) = -\text{csch}^2 x

\frac{d}{dx}(\operatorname{arcsinh} x) = \frac{1}{\sqrt{x^2 + 1}} \quad \frac{d}{dx}(\operatorname{arccosh} x) = \frac{1}{\sqrt{x-1}\sqrt{x+1}} \quad \frac{d}{dx}(\operatorname{arctanh} x) = \frac{1}{1 - x^2}

\frac{d}{dx}(\ln(ax+b)) = \frac{a}{ax+b} \quad \frac{d}{dx}(e^{kx}) = k e^{kx} \quad \frac{d}{dx}(\sin(kx)) = k\cos(kx) \quad \frac{d}{dx}(\cos(kx)) = -k\sin(kx) \\[6pt] \frac{d}{dx}(\tan(kx)) = k\sec^2(kx) \quad \frac{d}{dx}(x^n e^{kx}) = x^n k e^{kx} + n x^{n-1} e^{kx} \quad \frac{d}{dx}(a x^n + b x + c) = a n x^{n-1} + b

\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \quad \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}} \quad \frac{d}{dx}(f(g(x))) = f'(g(x)) g'(x) \\[6pt]

\frac{d}{dx}(C) = 0 \quad \frac{d}{dx}(ax + b) = a