derivative of {f}(x*1/({g)(x)})

Lösung

\frac{d}{d x} (f (x) \cdot \frac{1}{g ( x )})

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

Schritte zur Lösung

\frac{d}{d x} (f (x) \frac{1}{g ( x )})

Wende die Produktregel an:

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

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

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

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

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

Vereinfache

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

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

\frac{y ^{^{'}} - e ^{- t} + 6}{y} = - 6

2 t y^{^{'}} - y = 2 t y^{3} cos (t)

t an g e n t y = 3 x^{4}, (- 2, 48)

\frac{d}{d x} (t^{x})

\frac{d}{d x} (sec (\frac{π}{3}))