derivative of xln(xy)

Lösung

\frac{d}{d x} (x ln (x y))

ln (x y) + \frac{y + x \frac{d y}{d x}}{y}

Schritte zur Lösung

\frac{d}{d x} (x ln (x y))

Wende die Produktregel an:

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

f = x, g = ln (x y)

= \frac{d x}{d x} ln (x y) + \frac{d}{d x} (ln (x y)) x

\frac{d x}{d x} = 1

\frac{d}{d x} (ln (x y)) = \frac{y + x \frac{d y}{d x}}{x y}

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

Vereinfache

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

= ln (x y) + \frac{y + x \frac{d y}{d x}}{y}

x \to 0 lim (\frac{e ^{x} - 1}{13 x})

\frac{d y}{d t} = e^{4 t + 5 y}

\frac{d}{d x} (\frac{4 x ^{2} + 4 x + 8}{x})

\int 2 e^{t} d t

\frac{d y}{d x} = 18 - y