(\partial)/(\partial x)(2^{1/x})

Lösung

\frac{\partial}{\partial x} (2^{\frac{1}{x}})

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

Schritte zur Lösung

\frac{\partial}{\partial x} (2^{\frac{1}{x}})

Wende Exponentenregel an:

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

2^{\frac{1}{x}} = e^{\frac{1}{x} l n (2)}

= \frac{\partial}{\partial x} (e^{\frac{1}{x} l n (2)})

Wende die Kettenregel an:

e^{\frac{1}{x} l n (2)} \frac{\partial}{\partial x} (\frac{1}{x} ln (2))

= e^{\frac{1}{x} l n (2)} \frac{\partial}{\partial x} (\frac{1}{x} ln (2))

\frac{\partial}{\partial x} (\frac{1}{x} ln (2)) = - \frac{ln ( 2 )}{x ^{2}}

= e^{\frac{1}{x} l n (2)} (- \frac{ln ( 2 )}{x ^{2}})

Vereinfache

e^{\frac{1}{x} l n (2)} (- \frac{ln ( 2 )}{x ^{2}}) : - \frac{ln ( 2 ) \cdot 2 ^{\frac{1}{x}}}{x ^{2}}

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

s l o p e (3, 5), (8, 7)

\int \frac{3 x ^{2} - 6 x ^{2} + x}{x ^{3}} d x

d er i v a t i v e g (w) = 9 3 w

t an g e n t y = 5 x + 20, at x = 1

d er i v a t i v e f (x) = 16 x + 3