derivative f(x)=(x^2)/(7+sqrt(x))

Lösung

ableitung von

f (x) = \frac{x ^{2}}{7 + x}

\frac{4 x ( 7 + x ) - x ^{\frac{3}{2}}}{2 ( 7 + x ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{2}}{7 + x})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( x ^{2} ) ( 7 + x ) - \frac{d}{d x} ( 7 + x ) x ^{2}}{( 7 + x ) ^{2}}

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

\frac{d}{d x} (7 + x) = \frac{1}{2 x}

= \frac{2 x ( 7 + x ) - \frac{1}{2 x} x ^{2}}{( 7 + x ) ^{2}}

Vereinfache

\frac{2 x ( 7 + x ) - \frac{1}{2 x} x ^{2}}{( 7 + x ) ^{2}} : \frac{4 x ( 7 + x ) - x ^{\frac{3}{2}}}{2 ( 7 + x ) ^{2}}

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

x y^{^{'}} - 2 y = 2 y x^{2}

x \to 0 lim (x \cdot cot (2 x))

s l o p e y = x^{2} - 2 x - 3, (2, - 3)

\frac{\partial}{\partial x} (x y)

\frac{d}{d x} ((\frac{x}{y ^{2}}) - (\frac{y}{x ^{2}}))