人工智能教程 - 數學基礎課程1.1 - 數學分析（一）5-7 隱函數微分,指數,對數,導數的大公式

chain rule use

隱函數微分(Implicit differentiation)

EX:

${\color{Red} \frac{d}{dx}x^a = ax^{a-1}}$

so far,so good! a=0,a=+1/-1,a=+2/-2,…
implicit diff allows of any inverse
f’n provided we know the deriv of the function

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指數和對數

(exponentials and logarithms)

general :

${a^{x_{1}+x_{2}}=a^{x_{1}}.a^{x_{2}}}$

${(a^{x_{1}})^{x_2}=a^{x_{1}.{x_{2}}}}$

circuitous means, we’re taking a roundabout route.

${\color{Red} \frac{d}{dx} e^x = e^x}$

Natural log:

$y = e^x \Leftrightarrow lny =x$

$ln(x_1.x_2) = lnx_1+lnx_2$

$\frac{d}{dx} a^x = (lna).a^x$

General:

${\color{Red} (lnu)' = \frac{u'}{u}}$

${\color{Red} \lim_{n\rightarrow \infty }(1+\frac{1}{n}) ^n= e}$

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exponents:

$a_k=(1+\frac{1}{k})^k$

$\lim_{k\rightarrow \infty}a_k = e$

General:

${\color{Red} \frac{d}{dx}x^r = rx^{r-1}}$

two methods to verify

• base e
• log diff

Natural log is natural!

example of economics:

$\frac{p'}{p}=(lnp)'$

General:

${\color{Red} \frac{d}{dx}secx=secx.tanx}$

Read formula backwards:

${\color{Red} \lim _{\Delta x\rightarrow 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}}=f'(x)}$