derivative of {f}(x*e^{1/({f)(x)}})

Lösung

\frac{d}{d x} (f (x) \cdot e^{\frac{1}{f ( x )}})

e^{\frac{1}{f ( x )}} \frac{d}{d x} (f (x)) - \frac{e ^{\frac{1}{f ( x )}} \frac{d}{d x} ( f ( x ) )}{f ( x )}

Schritte zur Lösung

\frac{d}{d x} (f (x) e^{\frac{1}{f ( x )}})

Wende die Produktregel an:

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

f = f (x), g = e^{\frac{1}{f ( x )}}

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

\frac{d}{d x} (e^{\frac{1}{f ( x )}}) = - \frac{e ^{\frac{1}{f ( x )}} \frac{d}{d x} ( f ( x ) )}{f ( x ) ^{2}}

= \frac{d}{d x} (f (x)) e^{\frac{1}{f ( x )}} + (- \frac{e ^{\frac{1}{f ( x )}} \frac{d}{d x} ( f ( x ) )}{f ( x ) ^{2}}) f (x)

Vereinfache

\frac{d}{d x} (f (x)) e^{\frac{1}{f ( x )}} + (- \frac{e ^{\frac{1}{f ( x )}} \frac{d}{d x} ( f ( x ) )}{f ( x ) ^{2}}) f (x) : e^{\frac{1}{f ( x )}} \frac{d}{d x} (f (x)) - \frac{e ^{\frac{1}{f ( x )}} \frac{d}{d x} ( f ( x ) )}{f ( x )}

= e^{\frac{1}{f ( x )}} \frac{d}{d x} (f (x)) - \frac{e ^{\frac{1}{f ( x )}} \frac{d}{d x} ( f ( x ) )}{f ( x )}

d er i v a t i v e 6 (x + 2)^{\frac{1}{2}}

\int_{0}^{6} x^{2} 36 - x^{2} d x

y = \frac{3 x ^{5} - 7 x ^{2} - 4}{x ^{2}}

\frac{d}{d x} (4^{e^{x}})

d er i v a t i v e (3 x^{2} + 1) (2 x + 2)^{2}