derivative of x/(sin(x))

Lösung

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

csc (x) - x cot (x) csc (x)

Schritte zur Lösung

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

Vereinfache

\frac{x}{sin ( x )} : x csc (x)

= \frac{d}{d x} (x csc (x))

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = csc (x)

= \frac{d x}{d x} csc (x) + \frac{d}{d x} (csc (x)) x

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

\frac{d}{d x} (csc (x)) = - cot (x) csc (x)

= 1 \cdot csc (x) + (- cot (x) csc (x)) x

Vereinfache

= csc (x) - x cot (x) csc (x)