d/(dt)(({W}(t))(t-{W}(t))*{g}(t))

Lösung

\frac{d}{d t} ((W (t)) (t - W (t)) \cdot g (t))

t g (t) \frac{d}{d t} (W (t)) + t W (t) \frac{d}{d t} (g (t)) - 2 W (t) g (t) \frac{d}{d t} (W (t)) + W (t) g (t) - W (t)^{2} \frac{d}{d t} (g (t))

Schritte zur Lösung

\frac{d}{d t} ((W (t)) (t - W (t)) g (t))

Wende die Produktregel an:

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

f = W (t), g = (t - W (t)) g (t)

= \frac{d}{d t} (W (t)) (t - W (t)) g (t) + \frac{d}{d t} ((t - W (t)) g (t)) W (t)

\frac{d}{d t} ((t - W (t)) g (t)) = (1 - \frac{d}{d t} (W (t))) g (t) + \frac{d}{d t} (g (t)) (t - W (t))

= \frac{d}{d t} (W (t)) (t - W (t)) g (t) + ((1 - \frac{d}{d t} (W (t))) g (t) + \frac{d}{d t} (g (t)) (t - W (t))) W (t)

Vereinfache

\frac{d}{d t} (W (t)) (t - W (t)) g (t) + ((1 - \frac{d}{d t} (W (t))) g (t) + \frac{d}{d t} (g (t)) (t - W (t))) W (t) : t g (t) \frac{d}{d t} (W (t)) + t W (t) \frac{d}{d t} (g (t)) - 2 W (t) g (t) \frac{d}{d t} (W (t)) + W (t) g (t) - W (t)^{2} \frac{d}{d t} (g (t))

= t g (t) \frac{d}{d t} (W (t)) + t W (t) \frac{d}{d t} (g (t)) - 2 W (t) g (t) \frac{d}{d t} (W (t)) + W (t) g (t) - W (t)^{2} \frac{d}{d t} (g (t))

e x p an d e^{x y} (y) (x^{2} + y^{2}) + e^{x y} (y) (2 x)

e x p an d \frac{4 ( n + 1 ) ( 2 n + 1 )}{n ^{2}}

e x p an d 4 sin^{2} (2 t) + 4 cos^{2} (2 t) + (e^{t} - e^{- t})^{2}

e x p an d (x^{2} y^{3} - 4 y + 4 x - 3) d x + (3 x^{2} y^{2} - 4 x) d y

s im pl i f y 6 (1 - x - \frac{1 - x ^{2}}{2})