derivative of 5/(sqrt(x^2+16))

Lösung

\frac{d}{d x} (\frac{5}{x ^{2} + 16})

- \frac{5 x}{( x ^{2} + 16 ) ^{\frac{3}{2}}}

Schritte zur Lösung

\frac{d}{d x} (\frac{5}{x ^{2} + 16})

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 \frac{d}{d x} (\frac{1}{x ^{2} + 16})

Wende Exponentenregel an:

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

= 5 \frac{d}{d x} ((x^{2} + 16)^{- \frac{1}{2}})

Wende die Kettenregel an:

- \frac{1}{2 ( x ^{2} + 16 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + 16)

= - \frac{1}{2 ( x ^{2} + 16 ) ^{\frac{3}{2}}} \frac{d}{d x} (x^{2} + 16)

\frac{d}{d x} (x^{2} + 16) = 2 x

= 5 (- \frac{1}{2 ( x ^{2} + 16 ) ^{\frac{3}{2}}} \cdot 2 x)

Vereinfache

5 (- \frac{1}{2 ( x ^{2} + 16 ) ^{\frac{3}{2}}} \cdot 2 x) : - \frac{5 x}{( x ^{2} + 16 ) ^{\frac{3}{2}}}

= - \frac{5 x}{( x ^{2} + 16 ) ^{\frac{3}{2}}}