# 導數

(1)
\begin{align} f'(x_0)=\frac{\mathrm{d}f}{\mathrm{d}x}(x_0)=\lim_{\Delta x \to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x} \end{align}

(2)
\begin{align} f'(x)= \frac{dy}{dx} = \frac{d f(x)}{dx} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} \end{align}

# 可微分

(3)
\begin{align} f(c) = \lim_{\Delta x \to 0} f(c+\Delta x) = \lim_{\Delta x \to 0} f(c-\Delta x) \quad ; \quad \Delta x > 0 \end{align}

# 切線

（1）非垂直：過 ( a, f (a)) 且斜率為 f'(a) 之直線，若 f'(a) 存在；
（2）垂直：直線 x = a，且導數為無限大。

# 微分法則

(4)
\begin{eqnarray} && f'(c) &=& 0 \\ && f'(x^n) &=& n x^{n-1} \\ && f'(e^x) &=& e^x \\ && f'(a^x) &=& a^x ln(a)\\ && f'(ln(x)) &=& \frac{1}{x} \\ && f'(ln_a(x)) &=& \frac{1}{x ln(a)} \\ \end{eqnarray}

(5)
\begin{eqnarray} && f'(sin(x)) &=& cos(x) \\ && f'(cos(x)) &=& - sin(x) \\ && f'(tan(x)) &=& sec^2(x) \\ && f'(cot(x)) &=& - csc^2(x) \\ && f'(sec(x)) &=& tan(x) sec(x) \\ && f'(csc(x)) &=& - csc(x) cot(x) \\ \end{eqnarray}

(6)
\begin{eqnarray} \frac{d}{dx} sin(x) &=& \lim_{\Delta x \to 0} \frac{sin(x+\Delta x)-sin(x)}{\Delta x} \\ &=& \lim_{\Delta x \to 0} \frac{sin(x) cos(\Delta x)+cos(x) sin(\Delta x)-sin(x)}{\Delta x} \\ &=& \lim_{\Delta x \to 0} \frac{sin(x) (cos(\Delta x)-1)}{\Delta x} + \lim_{\Delta x \to 0} \frac{cos(x) (sin(\Delta x))}{\Delta x} \\ &=& sin(x) \lim_{\Delta x \to 0} \frac{ cos(\Delta x)-1}{\Delta x} + cos(x) \lim_{\Delta x \to 0} \frac{ sin(\Delta x)}{\Delta x}\\ &=& cos(x) \end{eqnarray}

(7)
\begin{eqnarray} && f'(sin^{-1}(x)) &=& \frac{1}{\sqrt{1-x^2}} && |x| < 1\\ && f'(cos^{-1}(x)) &=& \frac{-1}{\sqrt{1-x^2}} && |x| < 1 \\ && f'(tan^{-1}(x)) &=& \frac{1}{1+x^2} && x \in R \\ && f'(cot^{-1}(x)) &=& \frac{-1}{1+x^2} && x \in R \\ && f'(sec^{-1}(x)) &=& \frac{1}{|x| \sqrt{x^2-1}} && |x| > 1 \\ && f'(csc^{-1}(x)) &=& \frac{-1}{|x| \sqrt{x^2-1}} && |x| > 1 \\ \end{eqnarray}

(8)
\begin{eqnarray} && [c f(x)]' = c f'(x) \\ && [f(x)+g(x)]' = f'(x) + g'(x) \\ && [a f(x)+b g(x)]' = a f'(x) + b g'(x) \\ && [ f(x) g(x)]' = f(x) g'(x) + g(x) f'(x) \\ && [ \frac{f(x)}{g(x)}]' = \frac{g(x) f'(x) - f(x) g'(x)}{[g(x)^2]} \\ \end{eqnarray}

(9)
\begin{eqnarray} \frac{dy}{dx} = \frac{d f(u(x))}{dx} = \frac{d y(u)}{du} \frac{d u(x)}{dx} \end{eqnarray}

(10)
\begin{eqnarray} \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \end{eqnarray}

# 隱函數的微分

(11)
\begin{eqnarray} \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \end{eqnarray}

# 羅必達法則

(12)
\begin{eqnarray} 1.&\lim_{x \to a} f(x) \to 0 \; and \; \lim_{x \to a} g(x) \to 0 \\ 2.&\lim_{x \to a} f(x) \to \infty \; and\; \lim_{x \to a} g(x) \to \infty \\ \end{eqnarray}

(13)
\begin{align} \lim_{x \to a} \frac{f(x)}{g(x)}= \frac{f'(x)}{g'(x)} \end{align}