d/(dt)((x)(x-y+z+t-1))

Lösung

\frac{d}{d t} ((x) (x - y + z (t) + t - 1))

t \frac{d x}{d t} + 2 x \frac{d x}{d t} - y \frac{d x}{d t} + z \frac{d x}{d t} - \frac{d x}{d t} + x \frac{d z}{d t} + x - x \frac{d y}{d t}

Schritte zur Lösung

\frac{d}{d t} ((x) (x - y + z + t - 1))

Wende die Produktregel an:

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

f = x, g = x - y + z + t - 1

= \frac{d x}{d t} (x - y + z + t - 1) + \frac{d}{d t} (x - y + z + t - 1) x

\frac{d}{d t} (x - y + z + t - 1) = \frac{d x}{d t} + \frac{d z}{d t} + 1 - \frac{d y}{d t}

= \frac{d x}{d t} (x - y + z + t - 1) + (\frac{d x}{d t} + \frac{d z}{d t} + 1 - \frac{d y}{d t}) x

Vereinfache

\frac{d x}{d t} (x - y + z + t - 1) + (\frac{d x}{d t} + \frac{d z}{d t} + 1 - \frac{d y}{d t}) x : t \frac{d x}{d t} + 2 x \frac{d x}{d t} - y \frac{d x}{d t} + z \frac{d x}{d t} - \frac{d x}{d t} + x \frac{d z}{d t} + x - x \frac{d y}{d t}

= t \frac{d x}{d t} + 2 x \frac{d x}{d t} - y \frac{d x}{d t} + z \frac{d x}{d t} - \frac{d x}{d t} + x \frac{d z}{d t} + x - x \frac{d y}{d t}

e x p an d \frac{( 1 - 2 s )}{( s ^{2} + 4 s + 5 )}

e x p an d y^{(\frac{1}{3})} (y^{(\frac{2}{3})} + y^{(\frac{5}{3})})

e x p an d (3 x^{3} - 4 x - 2)^{6}

e x p an d \frac{( x - 1 ) ( x + 2 ) ( x + 3 )}{x ^{2}}

e x p an d - \frac{1}{4} (1 - s) (1 - t) + (- \frac{1}{4}) (1 + s) (1 - t)