basic

\( f'(a)= \) \( \lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{(a+h)-a} = \lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h} \)
\( \lim_{\Delta x\rightarrow 0} \frac{\Delta y}{\Delta x} \) \( = \lim_{x_2\rightarrow x_1} \frac{f(x_2)-f(x_1)}{x_2-x_1} \)

Leibniz notation

\( f'(a)= \) \( \frac{dy}{dx} |_{x=a} \) or \( \frac{dy}{dx} ]_{x=a} \)

fail to be differentiable

the product rule

\( (fg)'=fg'+f'g \)

the quotient rule

\( (\frac{f}{g})' = \frac{gf'-fg'}{g^2} \)

the power rule

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

trigonometric function

\( (\sin x )' = \cos x \)
\( (\cos x )' = -\sin x \)
\( (\tan x )' = \sec^2 x \)

\( (\cot x )' = -\csc^2x \)
\( (\sec x )' = \sec x \tan x \)
\( (\csc x )' = -\csc x \cot x \) 		hint:
\( \tan x = \frac{\sin x}{\cos x} = \frac{1}{\cot x} \)
\( \cot x = \frac{\cos x}{\sin x} = \frac{1}{\tan x} \)
\( \sec x = \frac{1}{\cos x} \)
\( \csc x = \frac{1}{\sin x} \)

the chain rule

\( \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} \)

\( \frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx} \)
\( = \frac{dy}{du} \frac{du}{dt} \frac{dt}{dx} \)

example:
\( f(x)=\sin(\cos(\tan x)) \)
\( f'(x)=\cos(\cos(\tan x))\frac{d}{dx}(\cos(\tan x)) \)
\( =\cos(\cos(\tan x))\cdot (-\sin(\tan x))\frac{d}{dx}(\tan x) \)
\( =\cos(\cos(\tan x))\cdot (-\sin(\tan x))\sec^2 x \)
\( (f\circ g)'(x) \)
\( =f'(g(x))\cdot g'(x) \)

***
\( f(x)=f(g(h(r(x)))) \)
\( f'(x)=f'(g(h(r(x)))) \times g'(h(r(x))) \times h'(r(x)) \times r'(x) \)

example:
\( \frac{d}{dx}\sin^2 x \)
\( \Rightarrow \text{suppose: } g(x)=\sin x \)
\( \Rightarrow \frac{d}{dx}(g(x))^2 \)
\( =2g(x)\cdot g'(x) = 2\sin x\cdot \cos x \)

example:(chain rule)
\( \frac{d}{dx}\sin(x^2) \)
the chain rule:
\( \text{Let }u=x^2, y=\sin(u) \)
\( \Rightarrow \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)
\( \Rightarrow \cos(u) \cdot \frac{du}{dx} \)
\( \Rightarrow \cos(x^2) \cdot 2x \)

linear approximations

wiki

\( L(x)= f(a)+f'(x)(x-a) \)

對數微分

reference