微積分上課筆記3

微積分上課筆記

微分

P.441

$\displaystyle (\ln x)’ = \frac{x’}{x}$ $\text{or}$ $\displaystyle [\ln(g(x))]’ = \frac{g’(x)}{g(x)}$

求$\displaystyle y=\frac{x^{\frac{3}{4}} \sqrt{x^2+1}}{(3x+2)^5}$之微分

$\displaystyle \frac{dy}{dx} \frac{1}{y}= \frac{3}{4} \frac{1}{x}+\frac{1}{2} \frac{2x}{x^2+1}-5\frac{3}{3x+2}$

$\ln e^x = x\ln e = x$
$(x\ln e)’ = (x)’ = 1$
$\displaystyle\frac{1}{e ^ 2} \frac{d}{dx}e^x=1$
$\displaystyle\rightarrow \frac{d}{dx}e^x=e^x$
唯一一個微分後不變的ㄈ

與$log$之關係

$\displaystyle \log_b^a=\frac{\ln a}{\ln b}$
if b is a constant
$\displaystyle

$\displaystyle (x^x)’ = e^{x \ln x}$

積分

求$\displaystyle \int_1^e \frac{\ln x}{x}dx$

$\displaystyle =\int_1^e (\ln x) \cdot d(\ln x)$
$\text{(due to}$ $(\ln x)’ = \frac{1}{x} \text{)}$
$\displaystyle =\frac{1}{2}(\ln x) ^ 2|_1^e$

反函數

$y = \ln x \rightarrow x = \exp y$

$\displaystyle e^x=y \rightarrow \ln y=x$
$\displaystyle e^{\ln x}=x$
$\ln{(e^x)} = x, x \in R$