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

Lösung

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

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

Schritte zur Lösung

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

Behandle

y

als Konstante

Entferne die Konstante:

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

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

Wende die Produktregel an:

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

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

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

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

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

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

Vereinfache

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

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

d er i v a t i v e ln (1 + e^{x})

t an g e n t \frac{5 x}{x + 4}

\int \frac{1}{4 x + x} d x

\frac{d}{d x} (\frac{3}{tan ( x )})

x \to 1 lim (\frac{∣ x + 1 ∣}{x + 1})