derivative of 4/((x+2^2+16))

Lösung

\frac{d}{d x} (\frac{4}{( x + 2 ) ^{2} + 16})

- \frac{4 ( 2 x + 4 )}{( x ^{2} + 4 x + 20 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{4}{( x + 2 ) ^{2} + 16})

Entferne die Konstante:

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

= 4 \frac{d}{d x} (\frac{1}{( x + 2 ) ^{2} + 16})

Wende Exponentenregel an:

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

= 4 \frac{d}{d x} (((x + 2)^{2} + 16)^{- 1})

Wende die Kettenregel an:

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

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

\frac{d}{d x} ((x + 2)^{2} + 16) = 2 x + 4

= 4 - \frac{1}{( ( x + 2 ) ^{2} + 16 ) ^{2}} (2 x + 4)

Vereinfache

4 - \frac{1}{( ( x + 2 ) ^{2} + 16 ) ^{2}} (2 x + 4) : - \frac{4 ( 2 x + 4 )}{( x ^{2} + 4 x + 20 ) ^{2}}

= - \frac{4 ( 2 x + 4 )}{( x ^{2} + 4 x + 20 ) ^{2}}