derivative of ((x+30sqrt(x^2+100))/x)

Lösung

\frac{d}{d x} (\frac{( x + 30 ) x ^{2} + 100}{x})

\frac{x ^{3} - 3000}{x ^{2} x ^{2} + 100}

Schritte zur Lösung

\frac{d}{d x} (\frac{( x + 30 ) x ^{2} + 100}{x})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( ( x + 30 ) x ^{2} + 100 ) x - \frac{d x}{d x} ( x + 30 ) x ^{2} + 100}{x ^{2}}

\frac{d}{d x} ((x + 30) x^{2} + 100) = \frac{2 x ^{2} + 30 x + 100}{x ^{2} + 100}

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

= \frac{\frac{2 x ^{2} + 30 x + 100}{x ^{2} + 100} x - 1 \cdot ( x + 30 ) x ^{2} + 100}{x ^{2}}

Vereinfache

\frac{\frac{2 x ^{2} + 30 x + 100}{x ^{2} + 100} x - 1 \cdot ( x + 30 ) x ^{2} + 100}{x ^{2}} : \frac{x ^{3} - 3000}{x ^{2} x ^{2} + 100}

= \frac{x ^{3} - 3000}{x ^{2} x ^{2} + 100}

d er i v a t i v e f (x) = 9 + 4 x

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