derivative of ((x^2+x(2x+1))/(4x))

Lösung

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

x + \frac{3}{4}

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} ((x^{2} + x) (2 x + 1)) = 6 x^{2} + 6 x + 1

\frac{d x}{d x} = 1

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

Vereinfache

\frac{1}{4} \cdot \frac{( 6 x ^{2} + 6 x + 1 ) x - 1 \cdot ( x ^{2} + x ) ( 2 x + 1 )}{x ^{2}} : x + \frac{3}{4}

= x + \frac{3}{4}

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