derivative of (x^2+sqrt(x)+1/(x^{3/2)})

Lösung

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

\frac{x ^{\frac{1}{2}} + x ^{2} - 3 - 3 x}{2 x ^{\frac{5}{2}}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (x^{2} + x + 1) = \frac{1 + 4 x ^{\frac{3}{2}}}{2 x ^{\frac{1}{2}}}

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

= \frac{\frac{1 + 4 x ^{\frac{3}{2}}}{2 x ^{\frac{1}{2}}} x ^{\frac{3}{2}} - \frac{3 x ^{\frac{1}{2}}}{2} ( x ^{2} + x + 1 )}{( x ^{\frac{3}{2}} ) ^{2}}

Vereinfache

\frac{\frac{1 + 4 x ^{\frac{3}{2}}}{2 x ^{\frac{1}{2}}} x ^{\frac{3}{2}} - \frac{3 x ^{\frac{1}{2}}}{2} ( x ^{2} + x + 1 )}{( x ^{\frac{3}{2}} ) ^{2}} : \frac{x ^{\frac{1}{2}} + x ^{2} - 3 - 3 x}{2 x ^{\frac{5}{2}}}

= \frac{x ^{\frac{1}{2}} + x ^{2} - 3 - 3 x}{2 x ^{\frac{5}{2}}}

s l o p e (- 4, - 3), (3, - 3)

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