derivative of ((x^2-1^{3/2})/(3x^3))

Lösung

\frac{d}{d x} \frac{( x ^{2} - 1 ) ^{\frac{3}{2}}}{3 x ^{3}}

\frac{( x ^{2} - 1 ) ^{\frac{1}{2}}}{x ^{4}}

Schritte zur Lösung

\frac{d}{d x} \frac{( x ^{2} - 1 ) ^{\frac{3}{2}}}{3 x ^{3}}

Entferne die Konstante:

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

= \frac{1}{3} \frac{d}{d x} \frac{( x ^{2} - 1 ) ^{\frac{3}{2}}}{x ^{3}}

Wende die Quotientenregel an:

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

= \frac{1}{3} \cdot \frac{\frac{d}{d x} ( ( x ^{2} - 1 ) ^{\frac{3}{2}} ) x ^{3} - \frac{d}{d x} ( x ^{3} ) ( x ^{2} - 1 ) ^{\frac{3}{2}}}{( x ^{3} ) ^{2}}

\frac{d}{d x} ((x^{2} - 1)^{\frac{3}{2}}) = 3 x (x^{2} - 1)^{\frac{1}{2}}

\frac{d}{d x} (x^{3}) = 3 x^{2}

= \frac{1}{3} \cdot \frac{3 x ( x ^{2} - 1 ) ^{\frac{1}{2}} x ^{3} - 3 x ^{2} ( x ^{2} - 1 ) ^{\frac{3}{2}}}{( x ^{3} ) ^{2}}

Vereinfache

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

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

x \to 9 + lim (\frac{2}{x - 9})

x \to 0 lim (\frac{a x ^{2} + b x ^{3}}{c x ^{2} + d x ^{3}})

d er i v a t i v e 5^{4 x}

\frac{d}{d x} ((7 x^{4} + 10)^{5})

x y^{^{'}} + 2 y - 6 = 0