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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Behandle

x

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

Beliebte Beispiele

integral of 2/(sqrt(4-x^2))

\int \frac{2}{4 - x ^{2}} d x

integral of ((x+2))/((x+1))

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

y^'(t)=(y+1)*(y-3),y(0)=1

y^{^{'}} (t) = (y + 1) \cdot (y - 3), y (0) = 1

integral of 1/2 (x+1/2 sin(2x))

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

limit as x approaches 1 of x^2-2x

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