derivative of (sin(x+cos(x))/(sin(x)))

Lösung

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

- csc^{2} (x)

Schritte zur Lösung

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

Vereinfache

\frac{sin ( x ) + cos ( x )}{sin ( x )} : 1 + cot (x)

= \frac{d}{d x} (1 + cot (x))

Wende die Summen-/Differenzregel an:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (1) + \frac{d}{d x} (cot (x))

\frac{d}{d x} (1) = 0

\frac{d}{d x} (cot (x)) = - csc^{2} (x)

= 0 - csc^{2} (x)

Vereinfache

= - csc^{2} (x)