derivative of xy^{-1}

Lösung

\frac{d}{d x} (x y^{- 1})

\frac{1}{y} - \frac{x \frac{d y}{d x}}{y ^{2}}

Schritte zur Lösung

\frac{d}{d x} (x y^{- 1})

Wende die Produktregel an:

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

f = x, g = y^{- 1}

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

\frac{d x}{d x} = 1

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

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

Vereinfache

1 \cdot y^{- 1} + (- \frac{\frac{d y}{d x}}{y ^{2}}) x : \frac{1}{y} - \frac{x \frac{d y}{d x}}{y ^{2}}

= \frac{1}{y} - \frac{x \frac{d y}{d x}}{y ^{2}}

\frac{\partial}{\partial x} (3 ln (\frac{2 x y}{5 z}))

\int \frac{cos ( t )}{2 + sin ( t )} d t

\frac{d}{d j} (3 i + 4 j)

\int 2 x^{2} cos (x^{3}) d x

\frac{d}{d x} (2 x y - 1)