derivative 1/(t^2+5t-1)

Lösung

ableitung von

\frac{1}{t ^{2} + 5 t - 1}

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

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

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

Vereinfache

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

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

t an g e n t y = \frac{4 x}{x ^{2} + 1}, (- 1, - 2)

a re a x^{3} - 13 x^{2} + 30 x, - x^{3} + 13 x^{2} - 30 x

l a pl a ce t r an s f or m - \frac{e ^{- 3 (t - 1)}}{3} - \frac{1}{3}

d er i v a t i v e f (x) = \frac{6 x ^{3}}{4 - x}

x \to 0 lim (\frac{3 ^{x^{2}} - 1}{x})