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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (cos (x^{2})) = - sin (x^{2}) \cdot 2 x

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

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

Vereinfache

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

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

\frac{\partial}{\partial x} (x^{0.5} y^{0.5} z)

(x + 2)^{2} y^{^{'}} + 8 y = 5 - 4 x y

x \to 4 lim (x^{2} - 5 x + 7 x^{3})

\frac{d}{d x} (\frac{1}{2 e ^{2}} e^{2 x})