derivative of x^x(ln(x+1))

Lösung

\frac{d}{d x} (x^{x} (ln (x) + 1))

x^{x} (ln (x) + 1)^{2} + x^{x - 1}

Schritte zur Lösung

\frac{d}{d x} (x^{x} (ln (x) + 1))

Wende die Produktregel an:

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

f = x^{x}, g = ln (x) + 1

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

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

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

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

Vereinfache

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

= x^{x} (ln (x) + 1)^{2} + x^{x - 1}

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

\frac{d}{d x} (x^{3} (x - 8)^{2})

\frac{d}{d x} (\frac{1}{x - y})

\frac{d y}{d x} + \frac{1}{x} y = 0

\int 2 cos (\frac{x}{3}) d x