derivative of (3cos(x)/(3x^2+3x-1))

Lösung

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

\frac{3 ( - sin ( x ) ( 3 x ^{2} + 3 x - 1 ) - cos ( x ) ( 6 x + 3 ) )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (cos (x)) = - sin (x)

\frac{d}{d x} (3 x^{2} + 3 x - 1) = 6 x + 3

= 3 \cdot \frac{( - sin ( x ) ) ( 3 x ^{2} + 3 x - 1 ) - ( 6 x + 3 ) cos ( x )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

Vereinfache

3 \cdot \frac{( - sin ( x ) ) ( 3 x ^{2} + 3 x - 1 ) - ( 6 x + 3 ) cos ( x )}{( 3 x ^{2} + 3 x - 1 ) ^{2}} : \frac{3 ( - sin ( x ) ( 3 x ^{2} + 3 x - 1 ) - cos ( x ) ( 6 x + 3 ) )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

= \frac{3 ( - sin ( x ) ( 3 x ^{2} + 3 x - 1 ) - cos ( x ) ( 6 x + 3 ) )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

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

\frac{d y}{d x} = \frac{x - y - 3}{x - 3}

in v erse l a pl a ce \frac{1}{0.012 s ^{2} + 1.55 s + 50}

s \to 1 lim (\frac{( 2 s + 3 )}{s})

\frac{d y}{d x} = 2 y + 3