derivative f(x)= 1/(sqrt(x^2+2x-5))

Lösung

ableitung von

f (x) = \frac{1}{x ^{2} + 2 x - 5}

- \frac{x + 1}{( x ^{2} + 2 x - 5 ) ^{\frac{3}{2}}}

Schritte zur Lösung

\frac{d}{d x} (\frac{1}{x ^{2} + 2 x - 5})

Wende Exponentenregel an:

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

= \frac{d}{d x} ((x^{2} + 2 x - 5)^{- \frac{1}{2}})

Wende die Kettenregel an:

- \frac{1}{2 ( x ^{2} + 2 x - 5 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + 2 x - 5)

= - \frac{1}{2 ( x ^{2} + 2 x - 5 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + 2 x - 5)

\frac{d}{d x} (x^{2} + 2 x - 5) = 2 x + 2

= - \frac{1}{2 ( x ^{2} + 2 x - 5 ) ^{\frac{3}{2}}} (2 x + 2)

Vereinfache

- \frac{1}{2 ( x ^{2} + 2 x - 5 ) ^{\frac{3}{2}}} (2 x + 2) : - \frac{x + 1}{( x ^{2} + 2 x - 5 ) ^{\frac{3}{2}}}

= - \frac{x + 1}{( x ^{2} + 2 x - 5 ) ^{\frac{3}{2}}}

\int_{0}^{1} \frac{1}{( 1 + x ^{2} ) ^{2}} d x

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