derivative of 5/(sqrt((7-x^2^5)))

Lösung

\frac{d}{d x} \frac{5}{( 7 - x ^{2} ) ^{5}}

\frac{25 x}{( 7 - x ^{2} ) ^{\frac{7}{2}}}

Schritte zur Lösung

\frac{d}{d x} \frac{5}{( 7 - x ^{2} ) ^{5}}

Entferne die Konstante:

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

= 5 \frac{d}{d x} \frac{1}{( 7 - x ^{2} ) ^{5}}

Wende Exponentenregel an:

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

= 5 \frac{d}{d x} (((7 - x^{2})^{5})^{- \frac{1}{2}})

Wende die Kettenregel an:

- \frac{1}{2 ( ( 7 - x ^{2} ) ^{5} ) ^{\frac{3}{2}}} \frac{d}{d x} ((7 - x^{2})^{5})

= - \frac{1}{2 ( ( 7 - x ^{2} ) ^{5} ) ^{\frac{3}{2}}} \frac{d}{d x} ((7 - x^{2})^{5})

\frac{d}{d x} ((7 - x^{2})^{5}) = - 10 x (7 - x^{2})^{4}

= 5 - \frac{1}{2 ( ( 7 - x ^{2} ) ^{5} ) ^{\frac{3}{2}}} (- 10 x (7 - x^{2})^{4})

Vereinfache

5 - \frac{1}{2 ( ( 7 - x ^{2} ) ^{5} ) ^{\frac{3}{2}}} (- 10 x (7 - x^{2})^{4}) : \frac{25 x}{( 7 - x ^{2} ) ^{\frac{7}{2}}}

= \frac{25 x}{( 7 - x ^{2} ) ^{\frac{7}{2}}}