(\partial)/(\partial y)(yln(x^2+y^3+5))

Lösung

\frac{\partial}{\partial y} (y ln (x^{2} + y^{3} + 5))

ln (x^{2} + y^{3} + 5) + \frac{3 y ^{3}}{x ^{2} + y ^{3} + 5}

Schritte zur Lösung

\frac{\partial}{\partial y} (y ln (x^{2} + y^{3} + 5))

Behandle

x

als Konstante

Wende die Produktregel an:

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

f = y, g = ln (x^{2} + y^{3} + 5)

= \frac{\partial}{\partial y} (y) ln (x^{2} + y^{3} + 5) + \frac{\partial}{\partial y} (ln (x^{2} + y^{3} + 5)) y

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

\frac{\partial}{\partial y} (ln (x^{2} + y^{3} + 5)) = \frac{3 y ^{2}}{x ^{2} + y ^{3} + 5}

= 1 \cdot ln (x^{2} + y^{3} + 5) + \frac{3 y ^{2}}{x ^{2} + y ^{3} + 5} y

Vereinfache

1 \cdot ln (x^{2} + y^{3} + 5) + \frac{3 y ^{2}}{x ^{2} + y ^{3} + 5} y : ln (x^{2} + y^{3} + 5) + \frac{3 y ^{3}}{x ^{2} + y ^{3} + 5}

= ln (x^{2} + y^{3} + 5) + \frac{3 y ^{3}}{x ^{2} + y ^{3} + 5}

\int d x d x

\frac{d}{d x} (\frac{( x ^{2} + 4 x + 3 )}{x})

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

n = 1 \sum \infty 8 (- 1)^{n} n x^{n}

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