derivative g(u)=sqrt(u+1)(1-8u^2)^7

Lösung

ableitung von

g (u) = u + 1 (1 - 8 u^{2})^{7}

\frac{( 1 - 8 u ^{2} ) ^{7}}{2 u + 1} - 112 u (1 - 8 u^{2})^{6} u + 1

Schritte zur Lösung

\frac{d}{d u} (u + 1 (1 - 8 u^{2})^{7})

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = u + 1, g = (1 - 8 u^{2})^{7}

= \frac{d}{d u} (u + 1) (1 - 8 u^{2})^{7} + \frac{d}{d u} ((1 - 8 u^{2})^{7}) u + 1

\frac{d}{d u} (u + 1) = \frac{1}{2 u + 1}

\frac{d}{d u} ((1 - 8 u^{2})^{7}) = - 112 u (1 - 8 u^{2})^{6}

= \frac{1}{2 u + 1} (1 - 8 u^{2})^{7} + (- 112 u (1 - 8 u^{2})^{6}) u + 1

Vereinfache

\frac{1}{2 u + 1} (1 - 8 u^{2})^{7} + (- 112 u (1 - 8 u^{2})^{6}) u + 1 : \frac{( 1 - 8 u ^{2} ) ^{7}}{2 u + 1} - 112 u (1 - 8 u^{2})^{6} u + 1

= \frac{( 1 - 8 u ^{2} ) ^{7}}{2 u + 1} - 112 u (1 - 8 u^{2})^{6} u + 1

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