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Beliebt Rechnen >

derivative of (1+1/x ^x)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende Exponentenregel an:

a^{b} = e^{b l n (a)}

(1 + \frac{1}{x})^{x} = e^{x l n (1 + \frac{1}{x})}

= \frac{d}{d x} (e^{x l n (1 + \frac{1}{x})})

Wende die Kettenregel an:

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

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

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

= e^{x l n (1 + \frac{1}{x})} (ln (1 + \frac{1}{x}) - \frac{1}{x + 1})

e^{x l n (1 + \frac{1}{x})} = (1 + \frac{1}{x})^{x}

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

Graph

Beliebte Beispiele

laplacetransform 0.025sin((2pit)/(10))

l a pl a ce t r an s f or m 0.025 sin (\frac{2 π t}{10})

derivative of x/((x-2^2))

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

integral from 1 to ln(8) of integral from 0 to ln(y) of e^{x+y}

\int_{1}^{l n (8)} \int_{0}^{l n (y)} e^{x + y} d x d y

integral of x/((x^2+1)^{1/2)}

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

fläche x^4-9x^2,x^2-9,0,3

a re a x^{4} - 9 x^{2}, x^{2} - 9, 0, 3