derivative of (x^3/(sin(x)))

Lösung

\frac{d}{d x} (\frac{x ^{3}}{sin ( x )})

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

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{3}}{sin ( x )})

Vereinfache

\frac{x ^{3}}{sin ( x )} : x^{3} csc (x)

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

Wende die Produktregel an:

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

f = x^{3}, g = csc (x)

= \frac{d}{d x} (x^{3}) csc (x) + \frac{d}{d x} (csc (x)) x^{3}

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

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

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

Vereinfache

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

\frac{( x ^{2} + 1 ) d y}{d x} + 7 x (y - 1) = 0

\int x \cdot (3 x + x) d x

\frac{\partial}{\partial x} (e^{4 x})

\int - \frac{1}{e ^{3 x}} d x

\int 5 cos^{3} (5 x) d x