derivative of 1/(f(x^2(x)))

Lösung

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

- \frac{3 f ( x ) + x \frac{d}{d x} ( f ( x ) )}{x ^{4} f ( x ) ^{2}}

Schritte zur Lösung

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

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

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

Wende die Kettenregel an:

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

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

\frac{d}{d x} (f (x) x^{2} x) = 3 x^{2} f (x) + \frac{d}{d x} (f (x)) x^{3}

= - \frac{1}{( f ( x ) x ^{2} x ) ^{2}} (3 x^{2} f (x) + \frac{d}{d x} (f (x)) x^{3})

Vereinfache

- \frac{1}{( f ( x ) x ^{2} x ) ^{2}} (3 x^{2} f (x) + \frac{d}{d x} (f (x)) x^{3}) : - \frac{3 f ( x ) + x \frac{d}{d x} ( f ( x ) )}{x ^{4} f ( x ) ^{2}}

= - \frac{3 f ( x ) + x \frac{d}{d x} ( f ( x ) )}{x ^{4} f ( x ) ^{2}}