Analisi matematica/Esempi derivata

[modifica] sviluppi di funzioni notevoli mediante la formula di Mac-Laurin

$e^{x}=1+\frac{x}{1!}+\frac{x^2}{2!}+....+\frac{x^{n-1}}{(n-1)!}+\frac{x^n}{n!}\ e^{\theta x}$
$a^x=1+\frac{x \log a}{1!}+\frac{(x \log a)^2}{2!}+...+\frac{(x \log a)^{n-1}}{(n-1)!}+\frac{(x \log a)^n}{n!}\ a^{\theta x}$
$sen\ x=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}+...+(-1)^k\ \frac{2^{2k+1}}{(2k+1)!}\ cos\ \theta x$
$cos\ x=1-{x^2\over 2!}+{x^4\over x^4!}-...+(-1)^k{x^{2k}\over2k!}cos\ \theta x$

forma $\frac{0}{0}\qquad \lim_{x\to\ -2}\frac{x^3+8}{x^5+32}=\frac{-8+8}{-32+32}=0$

$\lim_{x\to\ -2}\frac{3 x^2}{5 x^4}=\frac{3 (-2)^2}{5 (-2)^4}=\frac{3}{20};$

$\lim_{x\to\ 0}\frac{a^x-b^x}{c^x-d^x}=\frac{a^0-b^0}{c^0-d^0}=0$

$\lim_{x\to\ 0}\frac{a^x\log a-b^x\log b}{c^x\log c-d^x\log d}=\frac{\log a-\log b}{\log c-\log d};$

$\lim_{x\to\ 0}\frac{\log\frac{1-x^2}{1+x^2}}{sen\ x^2}=\frac{log\ 1}{sen\ 0^2}=\frac{0}{0}$

$\lim_{x\to\ 0}\frac{\log\frac{1-x^2}{1+x^2}}{sen\ x^2}=\lim_{x\to\ 0}{d\over dx}(\frac{1-x^2}{1+x^2}):{d\over dx}sen\ x^2=$

$\lim_{x\to\ 0}\frac{\frac{-4x}{1-x^4}}{2\ senx\ cosx}=\lim_{x\to\ 0}(-\frac{4}{2\ (1-x^4)\ cos\ x})\cdot\lim_{x\to\ 0}\frac{x}{sen\ x}=-2.$

forma $\frac{\infty}{\infty}\qquad \lim_{x\to\ 0}\frac{\frac{1}{log\ (1-x)}}{cotg\ x}=\frac{\frac{1}{log\ 1}}{cotg\ 0}=\frac{\infty}{\infty}$