Analisi matematica/Esempi derivata

Sviluppi di funzioni notevoli mediante la formula di Mac-Laurin

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

Calcolo del limite delle forme indeterminate

1. forma ${\frac {0}{0}}\qquad \lim _{x\to \ -2}{\frac {x^{3}+8}{x^{5}+32}}={\frac {-8+8}{-32+32}}=0$

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

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

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

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

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

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

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