(\partial)/(\partial y)(yarcsin(xy))

Lösung

\frac{\partial}{\partial y} (y arcsin (x y))

arcsin (y x) + \frac{y x}{1 - y ^{2} x ^{2}}

Schritte zur Lösung

\frac{\partial}{\partial y} (y arcsin (x y))

Behandle

x

als Konstante

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = y, g = arcsin (x y)

= \frac{\partial}{\partial y} (y) arcsin (x y) + \frac{\partial}{\partial y} (arcsin (x y)) y

\frac{\partial}{\partial y} (y) = 1

\frac{\partial}{\partial y} (arcsin (x y)) = \frac{x}{1 - x ^{2} y ^{2}}

= 1 \cdot arcsin (x y) + \frac{x}{1 - x ^{2} y ^{2}} y

Vereinfache

1 \cdot arcsin (x y) + \frac{x}{1 - x ^{2} y ^{2}} y : arcsin (y x) + \frac{y x}{1 - y ^{2} x ^{2}}

= arcsin (y x) + \frac{y x}{1 - y ^{2} x ^{2}}

d er i v a t i v e f (x) = 3 - x + x^{2}

t an g e n t \frac{x}{x + 5}

\frac{d}{d x} (- cos (x) + sin (x))

d er i v a t i v e f (x) = (3 x^{2} + 1)^{4}

t an g e n t 3 + 5 x^{2} - 2 x^{3}