derivative of {f}(x(x)^{-1})

Lösung

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

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

Schritte zur Lösung

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

Wende die Produktregel an:

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

f = f (x), g = x^{- 1}

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

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

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

Vereinfache

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

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

(x \frac{d y}{d x} + y) (2 (x + x y)) = x y

\frac{x y ^{^{'}}}{y - 2} = y^{2} - y

t an g e n t f (x) = \frac{- 6 x}{x ^{2} + 1}, (1, - 3)

a re a y = - x^{2} + 16, y = - \frac{1}{4} x - 1, 4 (y + 2.3) = 2.3 x

\int \frac{22 x ^{2}}{x ^{4} - 61 x ^{2} + 900} d x