derivative of 1/((x+y))

Lösung

\frac{d}{d x} (\frac{1}{( x + y )})

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

Schritte zur Lösung

\frac{d}{d x} (\frac{1}{( x + y )})

Wende Exponentenregel an:

\frac{1}{a} = a^{- 1}

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

x \to + 0 + lim ((1 + x)^{(\frac{1}{x})})

\frac{d y}{d x} = 1 + y - x

t an g e n t 3 x^{2} - 6 x - 9, at x = 3

\frac{\partial}{\partial y} (\frac{x y ^{2}}{x ^{2} y ^{3} + 1})

x \to 1 lim (\frac{x + 3}{x - 2})