d/(dy)((cos(x)+sin(y))^{ysin(2x-y)})

Lösung

\frac{d}{d y} ((cos (x) + sin (y))^{y s i n (2 x - y)})

(sin (2 x - y) ln (cos (x) + sin (y)) + y (- cos (2 x - y) ln (cos (x) + sin (y)) + \frac{sin ( 2 x - y ) cos ( y )}{cos ( x ) + sin ( y )})) (cos (x) + sin (y))^{y s i n (2 x - y)}

Schritte zur Lösung

\frac{d}{d y} ((cos (x) + sin (y))^{y s i n (2 x - y)})

Behandle

x

als Konstante

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

(cos (x) + sin (y))^{y s i n (2 x - y)} = e^{y s i n (2 x - y) l n (c o s (x) + s i n (y))}

= \frac{d}{d y} (e^{y s i n (2 x - y) l n (c o s (x) + s i n (y))})

Wende die Kettenregel an:

e^{y s i n (2 x - y) l n (c o s (x) + s i n (y))} \frac{d}{d y} (y sin (2 x - y) ln (cos (x) + sin (y)))

= e^{y s i n (2 x - y) l n (c o s (x) + s i n (y))} \frac{d}{d y} (y sin (2 x - y) ln (cos (x) + sin (y)))

\frac{d}{d y} (y sin (2 x - y) ln (cos (x) + sin (y))) = sin (2 x - y) ln (cos (x) + sin (y)) + y (- cos (2 x - y) ln (cos (x) + sin (y)) + \frac{sin ( 2 x - y ) cos ( y )}{cos ( x ) + sin ( y )})

= e^{y s i n (2 x - y) l n (c o s (x) + s i n (y))} (sin (2 x - y) ln (cos (x) + sin (y)) + y (- cos (2 x - y) ln (cos (x) + sin (y)) + \frac{sin ( 2 x - y ) cos ( y )}{cos ( x ) + sin ( y )}))

e^{y s i n (2 x - y) l n (c o s (x) + s i n (y))} = (cos (x) + sin (y))^{y s i n (2 x - y)}

= (sin (2 x - y) ln (cos (x) + sin (y)) + y (- cos (2 x - y) ln (cos (x) + sin (y)) + \frac{sin ( 2 x - y ) cos ( y )}{cos ( x ) + sin ( y )})) (cos (x) + sin (y))^{y s i n (2 x - y)}

t a y l or 2 x ln (x + 1)

d er i v a t i v e y = \frac{200}{0.9 + 0.1 e ^{2 x}}

x \to 0 lim (\frac{2}{x ^{2} + x})

\int \frac{( 2 x ^{2} + 3 x + 2 )}{( x ^{2} + 1 ) ^{2}} d x

\frac{\partial}{\partial v} (2 v cos (u))