derivative of (5x-x^2/(sqrt(x))-3cot(x))

Lösung

\frac{d}{d x} (\frac{5 x - x ^{2}}{x} - 3 cot (x))

\frac{5 - 3 x}{2 x} + 3 csc^{2} (x)

Schritte zur Lösung

\frac{d}{d x} (\frac{5 x - x ^{2}}{x} - 3 cot (x))

Vereinfache

\frac{5 x - x ^{2}}{x} - 3 cot (x) : x (5 - x) - 3 cot (x)

= \frac{d}{d x} (x (5 - x) - 3 cot (x))

Wende die Summen-/Differenzregel an:

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

= \frac{d}{d x} (x (5 - x)) - \frac{d}{d x} (3 cot (x))

\frac{d}{d x} (x (5 - x)) = \frac{5 - 3 x}{2 x}

\frac{d}{d x} (3 cot (x)) = - 3 csc^{2} (x)

= \frac{5 - 3 x}{2 x} - (- 3 csc^{2} (x))

Vereinfache

= \frac{5 - 3 x}{2 x} + 3 csc^{2} (x)