derivative of (cos(x)/(sin^2(x)))

Lösung

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

csc (x) - 2 csc^{3} (x)

Schritte zur Lösung

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

Vereinfache

\frac{cos ( x )}{sin ^{2} ( x )} : cot (x) csc (x)

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

Wende die Produktregel an:

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

f = cot (x), g = csc (x)

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

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

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

= (- csc^{2} (x)) csc (x) + (- cot (x) csc (x)) cot (x)

Vereinfache

(- csc^{2} (x)) csc (x) + (- cot (x) csc (x)) cot (x) : csc (x) - 2 csc^{3} (x)

= csc (x) - 2 csc^{3} (x)

x \to 0 lim (x \cdot e^{\frac{1}{x}} - 1)

\frac{d y}{d x} = \frac{y ( ln ( 5 y ) - ln ( 5 x ) + 7 )}{7 x}

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

\int_{0}^{2.5} 25 - x^{2} d x

\int \frac{1}{x ^{2} + 14} d x