(\partial)/(\partial y)((x^2+y^2)ln(x^2+y^2))

Lösung

\frac{\partial}{\partial y} ((x^{2} + y^{2}) ln (x^{2} + y^{2}))

2 y ln (x^{2} + y^{2}) + 2 y

Schritte zur Lösung

\frac{\partial}{\partial y} ((x^{2} + y^{2}) ln (x^{2} + y^{2}))

Behandle

x

als Konstante

Wende die Produktregel an:

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

f = x^{2} + y^{2}, g = ln (x^{2} + y^{2})

= \frac{\partial}{\partial y} (x^{2} + y^{2}) ln (x^{2} + y^{2}) + \frac{\partial}{\partial y} (ln (x^{2} + y^{2})) (x^{2} + y^{2})

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

\frac{\partial}{\partial y} (ln (x^{2} + y^{2})) = \frac{2 y}{x ^{2} + y ^{2}}

= 2 y ln (x^{2} + y^{2}) + \frac{2 y}{x ^{2} + y ^{2}} (x^{2} + y^{2})

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

= 2 y ln (x^{2} + y^{2}) + 2 y

d er i v a t i v e 5 t^{- \frac{3}{5}}

t a y l or \frac{5}{3 - 2 x}

x \to - 2 lim (x^{2} + 1)

t an g e n t \frac{6 x}{x ^{2} + 1}

x \to + 0 lim ((3 x)^{x^{2}})