cot u türevi.png

¿ Cuál es la Derivada de Cot u ?

La derivada de cot u es -u'.(1+cot² u).

$(co t u)^{'} = - u^{'} . (1 + co t^{2} u)$

$\frac{d}{d x} (co t u) = - u^{'} . (1 + co t^{2} u)$

Prueba de la Derivada de Cot u

$f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}$

$[co t u (x)]^{'} = h \to 0 lim \frac{co t u ( x + h ) - co t u ( x )}{h}$

$co t x = \frac{cos x}{s in x}$

$[co t u (x)]^{'} = h \to 0 lim \frac{\frac{cos u ( x + h )}{s in u ( x + h )} - \frac{cos u ( x )}{s in u ( x )}}{h}$

$[co t u (x)]^{'} = h \to 0 lim \frac{\frac{s in u ( x ) . cos u ( x + h ) - cos u ( x ) . s in u ( x + h )}{s in u ( x ) . s in u ( x + h )}}{h}$

$[co t u (x)]^{'} = h \to 0 lim \frac{s in u ( x ) . cos u ( x + h ) - cos u ( x ) . s in u ( x + h )}{h . s in u ( x ) . s in u ( x + h )}$

$s in p . cos q - cos p . s in q = s in (p - q)$

$[co t u (x)]^{'} = h \to 0 lim \frac{s in [ u ( x ) - u ( x + h )]}{h . s in u ( x ) . s in u ( x + h )}$

$[co t u (x)]^{'} = h \to 0 lim \frac{s in { - [ u ( x + h ) - u ( x )]}}{h . s in u ( x ) . s in u ( x + h )}$

$s in (- x) = - s in x$

$[co t u (x)]^{'} = h \to 0 lim \frac{- s in [ u ( x + h ) - u ( x )]}{h . s in u ( x ) . s in u ( x + h )}$

$[co t u (x)]^{'} = h \to 0 lim \frac{- [ u ( x + h ) - u ( x )] . s in [ u ( x + h ) - u ( x )]}{[ u ( x + h ) - u ( x )] . h . s in u ( x ) . s in u ( x + h )}$

$[co t u (x)]^{'} = h \to 0 lim {\frac{- [ u ( x + h ) - u ( x )]}{h} . \frac{s in [ u ( x + h ) - u ( x )]}{u ( x + h ) - u ( x )} . \frac{1}{s in u ( x ) . s in u ( x + h )}}$

$[co t u (x)]^{'} = h \to 0 lim \frac{- [ u ( x + h ) - u ( x )]}{h} . h \to 0 lim \frac{s in [ u ( x + h ) - u ( x )]}{u ( x + h ) - u ( x )} . h \to 0 lim \frac{1}{s in u ( x ) . s in u ( x + h )}$

$[co t u (x)]^{'} = - h \to 0 lim \frac{u ( x + h ) - u ( x )}{h} . h \to 0 lim \frac{s in [ u ( x + h ) - u ( x )]}{u ( x + h ) - u ( x )} . h \to 0 lim \frac{1}{s in u ( x ) . s in u ( x + h )}$

$h \to 0 [u (x + h) - u (x) = h]$

$[co t u (x)]^{'} = - h \to 0 lim \frac{u ( x + h ) - u ( x )}{h} . h \to 0 lim \frac{s in h}{h} . h \to 0 lim \frac{1}{s in u ( x ) . s in u ( x + h )}$

$t \to 0 l i m \frac{s in t}{t} = 1$

$[co t u (x)]^{'} = - u^{'} (x) .1. \frac{1}{s in u ( x ) . s in u ( x + 0 )}$

$[co t u (x)]^{'} = - u^{'} (x) .1. \frac{1}{s in u ( x ) . s in u ( x )}$

$[co t u (x)]^{'} = - \frac{u ^{'} ( x ) .1.1}{s i n ^{2} u ( x )}$

$[co t u (x)]^{'} = - \frac{u ^{'} ( x )}{s i n ^{2} u ( x )}$

$u (x) = u$

$u^{'} (x) = u^{'}$

$(co t u)^{'} = - \frac{u ^{'}}{s i n ^{2} u}$

$(co t u)^{'} = - u^{'} . \frac{1}{s i n ^{2} u}$

$(co t u)^{'} = - u^{'} . (\frac{1}{s in u})^{2}$

$\frac{1}{s in x} = csc x$

$(co t u)^{'} = - u^{'} . cs c^{2} u$

$(co t u)^{'} = - u^{'} . \frac{1}{s i n ^{2} u}$

$1 = s i n^{2} x + co s^{2} x$

$(co t u)^{'} = - u^{'} . (\frac{s i n ^{2} u + co s ^{2} u}{s i n ^{2} u})$

$(co t u)^{'} = - u^{'} . (\frac{s i n ^{2} u}{s i n ^{2} u} + \frac{co s ^{2} u}{s i n ^{2} u})$

$(co t u)^{'} = - u^{'} . [1 + (\frac{cos u}{s in u})^{2}]$

$(co t u)^{'} = - u^{'} . (1 + co t^{2} u)$

$(co t u)^{'} = - u^{'} . (1 + co t^{2} u) = - \frac{u ^{'}}{s i n ^{2} u} = - u^{'} . cs c^{2} u$

Pregunta

$f (x) = co t 3 x \Rightarrow f^{'} (x) = ?$

Respuesta

$(co t u)^{'} = - u^{'} . (1 + co t^{2} u)$

$(co t 3 x)^{'} = - (3 x)^{'} . (1 + co t^{2} 3 x)$

$(co t 3 x)^{'} = - 3. (1 + co t^{2} 3 x)$

$(co t 3 x)^{'} = - 3.1 - 3. co t^{2} 3 x$

$(co t 3 x)^{'} = - 3 - 3. co t^{2} 3 x$

Derivada de Cot uself.__wrap_n!=1&&self.__wrap_b(":R35nlttada:",1)

Derivada de Cot u