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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

\frac{d x}{d x} = 1

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

Vereinfache

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

- y^{^{''}} + y = 1

a re a f (x) = - x^{2} + 3, g (x) = 2 x + 3

x \to - 7 lim (x)

\frac{d}{d x} (5 x - tan (5 x))

y^{^{'}} + y = 5 sin (2 t)