derivative of 1/(sqrt(x^2+2y^2-5))

Lösung

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

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

Schritte zur Lösung

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

Wende Exponentenregel an:

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

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

Wende die Kettenregel an:

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

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

\frac{d}{d x} (x^{2} + 2 y^{2} - 5) = 2 x + 4 y \frac{d y}{d x}

= - \frac{1}{2 ( x ^{2} + 2 y ^{2} - 5 ) ^{\frac{3}{2}}} (2 x + 4 y \frac{d y}{d x})

Vereinfache

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

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

d er i v a t i v e f (x) = 5 x + 6 x

d er i v a t i v e f (x) = 20 x

x \to 0 lim (\frac{sin ( 13 x )}{x})

\int \frac{2900}{1 + 0.25 x} d x

t an g e n t f (x) = \frac{1}{4 x}, at x = 9