d/(dt)(({z}(t,x,y))(sqrt(1+x^2)+y^2))

Lösung

\frac{d}{d t} ((z (t, x, y)) (1 + x^{2} + y^{2}))

\frac{d}{d t} (z (t, x, y)) (1 + x^{2} + y^{2}) + (\frac{x \frac{d x}{d t}}{1 + x ^{2}} + 2 y \frac{d y}{d t}) z (t, x, y)

Schritte zur Lösung

\frac{d}{d t} ((z (t, x, y)) (1 + x^{2} + y^{2}))

Wende die Produktregel an:

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

f = z (t, x, y), g = 1 + x^{2} + y^{2}

= \frac{d}{d t} (z (t, x, y)) (1 + x^{2} + y^{2}) + \frac{d}{d t} (1 + x^{2} + y^{2}) z (t, x, y)

\frac{d}{d t} (1 + x^{2} + y^{2}) = \frac{x \frac{d x}{d t}}{1 + x ^{2}} + 2 y \frac{d y}{d t}

= \frac{d}{d t} (z (t, x, y)) (1 + x^{2} + y^{2}) + (\frac{x \frac{d x}{d t}}{1 + x ^{2}} + 2 y \frac{d y}{d t}) z (t, x, y)

e x p an d - 8 (1 x + 7 y - 6 z = - 7)

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

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

e x p an d x (1, 50 + 3 + 4.5 + 3)

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