d/(dt)(({z}(t,x,y))(x^2+y^5))

Lösung

\frac{d}{d t} ((z (t, x, y)) (x^{2} + y^{5}))

\frac{d}{d t} (z (t, x, y)) (x^{2} + y^{5}) + (2 x \frac{d x}{d t} + 5 y^{4} \frac{d y}{d t}) z (t, x, y)

Schritte zur Lösung

\frac{d}{d t} ((z (t, x, y)) (x^{2} + y^{5}))

Wende die Produktregel an:

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

f = z (t, x, y), g = x^{2} + y^{5}

= \frac{d}{d t} (z (t, x, y)) (x^{2} + y^{5}) + \frac{d}{d t} (x^{2} + y^{5}) z (t, x, y)

\frac{d}{d t} (x^{2} + y^{5}) = 2 x \frac{d x}{d t} + 5 y^{4} \frac{d y}{d t}

= \frac{d}{d t} (z (t, x, y)) (x^{2} + y^{5}) + (2 x \frac{d x}{d t} + 5 y^{4} \frac{d y}{d t}) z (t, x, y)

e x p an d \frac{9 x - 8}{( x - 3 ) ( 2 x - 5 )}

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

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

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

e x p an d 5 y^{3} d x - y^{2} (- 2 x + y^{2} z^{4}) d y