(\partial)/(\partial y)(x^3y^{10}sin(xy^2))

Lösung

\frac{\partial}{\partial y} (x^{3} y^{10} sin (x y^{2}))

x^{3} (10 y^{9} sin (x y^{2}) + 2 x y^{11} cos (x y^{2}))

Schritte zur Lösung

\frac{\partial}{\partial y} (x^{3} y^{10} sin (x y^{2}))

Behandle

x

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= x^{3} \frac{\partial}{\partial y} (y^{10} sin (x y^{2}))

Wende die Produktregel an:

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

f = y^{10}, g = sin (x y^{2})

= x^{3} (\frac{\partial}{\partial y} (y^{10}) sin (x y^{2}) + \frac{\partial}{\partial y} (sin (x y^{2})) y^{10})

\frac{\partial}{\partial y} (y^{10}) = 10 y^{9}

\frac{\partial}{\partial y} (sin (x y^{2})) = cos (x y^{2}) \cdot 2 x y

= x^{3} (10 y^{9} sin (x y^{2}) + cos (x y^{2}) \cdot 2 x y y^{10})

cos (x y^{2}) \cdot 2 x y y^{10} = 2 x y^{11} cos (x y^{2})

= x^{3} (10 y^{9} sin (x y^{2}) + 2 x y^{11} cos (x y^{2}))

\int 6 t e^{3 t} d t

x \to 0 lim (\frac{x ^{3}}{x ^{3}})

\frac{d}{d x} (a^{2} sin^{2} (x))

t an g e n t f (x) = \frac{x}{x + 3}, at x = - 2

t \to + 0 lim (\frac{sin ( 2 t )}{t})