derivative of (5x^2/(2sqrt(b^2+x^2)))

Lösung

\frac{d}{d x} (\frac{5 x ^{2}}{2 b ^{2} + x ^{2}})

\frac{5 ( 2 b ^{2} x + x ^{3} )}{2 ( b ^{2} + x ^{2} ) b ^{2} + x ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{5 x ^{2}}{2 b ^{2} + x ^{2}})

Entferne die Konstante:

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

= \frac{5}{2} \frac{d}{d x} (\frac{x ^{2}}{b ^{2} + x ^{2}})

Wende die Quotientenregel an:

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

= \frac{5}{2} \cdot \frac{\frac{d}{d x} ( x ^{2} ) b ^{2} + x ^{2} - \frac{d}{d x} ( b ^{2} + x ^{2} ) x ^{2}}{( b ^{2} + x ^{2} ) ^{2}}

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

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

= \frac{5}{2} \cdot \frac{2 x b ^{2} + x ^{2} - \frac{x}{b ^{2} + x ^{2}} x ^{2}}{( b ^{2} + x ^{2} ) ^{2}}

Vereinfache

\frac{5}{2} \cdot \frac{2 x b ^{2} + x ^{2} - \frac{x}{b ^{2} + x ^{2}} x ^{2}}{( b ^{2} + x ^{2} ) ^{2}} : \frac{5 ( 2 b ^{2} x + x ^{3} )}{2 ( b ^{2} + x ^{2} ) b ^{2} + x ^{2}}

= \frac{5 ( 2 b ^{2} x + x ^{3} )}{2 ( b ^{2} + x ^{2} ) b ^{2} + x ^{2}}

(y^{2} + x y) d x + x^{2} d y = 0

20 y^{^{''}} + 11 y^{^{'}} - 3 y = 0

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

x \to 1 lim (\frac{x ^{2} - 3 x + 2}{x ^{3} - 1})

\int \frac{3}{x ^{2} 25 - x ^{2}} d x