derivative of (sin(pi/2 x)/x)

Lösung

\frac{d}{d x} (\frac{sin ( \frac{π}{2} x )}{x})

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

Schritte zur Lösung

\frac{d}{d x} (\frac{sin ( \frac{π}{2} x )}{x})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (sin (\frac{π}{2} x)) = \frac{π}{2} cos (\frac{π x}{2})

\frac{d x}{d x} = 1

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

Vereinfache

\frac{\frac{π}{2} cos ( \frac{π x}{2} ) x - 1 \cdot sin ( \frac{π}{2} x )}{x ^{2}} : \frac{π x cos ( \frac{π x}{2} ) - 2 sin ( \frac{π x}{2} )}{2 x ^{2}}

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

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

\int_{0}^{\frac{1}{5}} 5 arcsin (5 y) d y

f (x) = cos (5 x)

x \to - 1 lim (x + 2)

\frac{\partial}{\partial x} (18 x y)