derivative of (x^{7/3}+3x^3-8/(6x^2))

Lösung

\frac{d}{d x} (\frac{x ^{\frac{7}{3}} + 3 x ^{3} - 8}{6 x ^{2}})

\frac{9 x ^{3} + x ^{\frac{7}{3}} + 48}{18 x ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{\frac{7}{3}} + 3 x ^{3} - 8}{6 x ^{2}})

Entferne die Konstante:

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

= \frac{1}{6} \frac{d}{d x} (\frac{x ^{\frac{7}{3}} + 3 x ^{3} - 8}{x ^{2}})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{6} \cdot \frac{\frac{d}{d x} ( x ^{\frac{7}{3}} + 3 x ^{3} - 8 ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( x ^{\frac{7}{3}} + 3 x ^{3} - 8 )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (x^{\frac{7}{3}} + 3 x^{3} - 8) = 9 x^{2} + \frac{7 x ^{\frac{4}{3}}}{3}

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

= \frac{1}{6} \cdot \frac{( 9 x ^{2} + \frac{7 x ^{\frac{4}{3}}}{3} ) x ^{2} - 2 x ( x ^{\frac{7}{3}} + 3 x ^{3} - 8 )}{( x ^{2} ) ^{2}}

Vereinfache

\frac{1}{6} \cdot \frac{( 9 x ^{2} + \frac{7 x ^{\frac{4}{3}}}{3} ) x ^{2} - 2 x ( x ^{\frac{7}{3}} + 3 x ^{3} - 8 )}{( x ^{2} ) ^{2}} : \frac{9 x ^{3} + x ^{\frac{7}{3}} + 48}{18 x ^{3}}

= \frac{9 x ^{3} + x ^{\frac{7}{3}} + 48}{18 x ^{3}}

x \to 0 lim (\frac{x + tan ( x )}{x})

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