derivative of (x^4-5x^3+sqrt(x)/(x^2))

Lösung

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

\frac{4 x ^{4} - 10 x ^{3} - 3 x}{2 x ^{3}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( x ^{4} - 5 x ^{3} + x ) x ^{2} - \frac{d}{d x} ( x ^{2} ) ( x ^{4} - 5 x ^{3} + x )}{( x ^{2} ) ^{2}}

\frac{d}{d x} (x^{4} - 5 x^{3} + x) = 4 x^{3} - 15 x^{2} + \frac{1}{2 x}

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

= \frac{( 4 x ^{3} - 15 x ^{2} + \frac{1}{2 x} ) x ^{2} - 2 x ( x ^{4} - 5 x ^{3} + x )}{( x ^{2} ) ^{2}}

Vereinfache

\frac{( 4 x ^{3} - 15 x ^{2} + \frac{1}{2 x} ) x ^{2} - 2 x ( x ^{4} - 5 x ^{3} + x )}{( x ^{2} ) ^{2}} : \frac{4 x ^{4} - 10 x ^{3} - 3 x}{2 x ^{3}}

= \frac{4 x ^{4} - 10 x ^{3} - 3 x}{2 x ^{3}}

\frac{d y}{d x} = \frac{e ^{x} + 1}{2 y}

x \to - \infty lim (4 x)

\int \frac{1}{3 - 2 x} d x

\int \frac{1}{1 + 4 x ^{2}} d x

\frac{d y}{d x} + x y^{2} = 0