Chapter 2 Rules of Differentiation(2/2)

Suppose $y = f(x) = 3x+5$
$\Rightarrow x = \frac{1}{3}(y-5)$
$\Rightarrow y = \frac{1}{3}(x-5)$
$f^{-1}(x) = \frac{1}{3}(x-5)$ is the inverse function of $f(x) = 3x+5$

Consider $y = f(x) = e^x$
$\Rightarrow log_ey = log_e e^x = x$
$\Rightarrow x = log_ey$
$\Rightarrow y = log_ex$
$f^{-1}(x) = log_ex$ is the inverse function of $f(x) = e^x$

1. $f(x) = sin^{-1}x \; \Rightarrow \; f:[-1:1]\to [ -\frac{\pi}{2},\frac{\pi}{2}]$
2. $f(x) = cos^{-1}x \; \Rightarrow \; f:[-1:1]\to [ 0 , \pi ]$
3. $f(x) = tan^{-1}x \; \Rightarrow \; f: \mathbb{R} \to [ -\frac{\pi}{2},\frac{\pi}{2}]$
4. $f(x) = cot^{-1}x \; \Rightarrow \; f: \mathbb{R} \to [ 0 , \pi ]$
5. $f(x) = sec^{-1}x \; \Rightarrow \; f: (-\infty,-1] \cup [1,\infty) \to [ 0 , \frac{\pi}{2}) \cup (\frac{\pi}{2},\pi]$
6. $f(x) = csc^{-1}x \; \Rightarrow \; f: (-\infty,-1] \cup [1,\infty) \to [ -\frac{\pi}{2} , 0 ) \cup ( 0, \frac{\pi}{2}]$

Let $f(x) = \frac{1}{4}x^3+x-1$

1. What is the value of $f^{-1}(x)$ when $x = 3$ ?
2. What is the value of $(f^{-1})'(x)$ when $x = 3$ ?

Let $f(x) = 3 \Rightarrow x^3 + 4x - 16 = 0$
$\Rightarrow (x-2)(x^2+2x+8) = 0 \therefore x = 2 \Rightarrow f^{-1}(3) = 2$

$\frac{d}{dx}|_{x=3}f^{-1}(x) = \frac{1}{\frac{d}{dy}|_{y=f^{-1}(3)}f(y)} = \frac{1}{(\frac{3}{4}y^2+1)|_{y=2}} = \frac{1}{4}$.

1. 反三角函數
2. 指對數函數

$y = f^{-1}(x) = sin^{-1}x$
$\frac{d}{dx} f^{-1}(x) = \frac{d}{dx} sin^{-1}x = \frac{1}{\frac{d}{dy}f(y)}=\frac{1}{cosy} = \frac{1}{\sqrt{1-x^2}}$

1. $\frac{d}{dx}e^x = e^x$
2. $\frac{d}{dx}a^x = (lna) \cdot a^x$
3. $\frac{d}{dx}(log_ax) = \frac{1}{\frac{d}{dy}a^y} = \frac{1}{(lna)a^y} = \frac{1}{xlna}$
   $\frac{d}{dx}lnx = \frac{1}{x},x>0$
4. $\frac{d}{dx}x^n = nx^{n-1}$if $n\in \mathbb{R}$
5. Let $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$ Then $e^x = \lim_{n\to \infty}(1+\frac{x}{n})^n$