d/(dy)((x(y))((y^4)/8+1/(4y^2)))

Lösung

\frac{d}{d y} ((x (y)) (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}))

\frac{x ( 5 y ^{6} - 2 )}{8 y ^{2}}

Schritte zur Lösung

\frac{d}{d y} ((x (y)) (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}))

Behandle

x

als Konstante

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= x \frac{d}{d y} (y (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}))

Wende die Produktregel an:

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

f = y, g = \frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}

= x (\frac{d y}{d y} (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}) + \frac{d}{d y} (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}) y)

\frac{d y}{d y} = 1

\frac{d}{d y} (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}) = \frac{y ^{3}}{2} - \frac{1}{2 y ^{3}}

= x (1 \cdot (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}) + (\frac{y ^{3}}{2} - \frac{1}{2 y ^{3}}) y)

Vereinfache

x (1 \cdot (\frac{y ^{4}}{8} + \frac{1}{4 y ^{2}}) + (\frac{y ^{3}}{2} - \frac{1}{2 y ^{3}}) y) : \frac{x ( 5 y ^{6} - 2 )}{8 y ^{2}}

= \frac{x ( 5 y ^{6} - 2 )}{8 y ^{2}}

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

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

e x p an d (- 1, 3) P (4, - 2)

s im pl i f y (\frac{3}{2} - x) (\frac{11}{2} - x)

s im pl i f y (2 x + \frac{1}{2}) (4 x^{2} - x + \frac{1}{4})