derivative of ((x^2+2x/(sin(x)))^3)

解答

\frac{d}{d x} ((\frac{x ^{2} + 2 x}{sin ( x )})^{3})

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

求解步骤

\frac{d}{d x} ((\frac{x ^{2} + 2 x}{sin ( x )})^{3})

化简

(\frac{x ^{2} + 2 x}{sin ( x )})^{3} : csc^{3} (x) (x^{2} + 2 x)^{3}

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

使用微分乘法定则:

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

f = csc^{3} (x), g = (x^{2} + 2 x)^{3}

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

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

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

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

化简

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

\int x^{- 13} d x

\int \frac{1}{x ^{2} x ^{2} - 49} d x

\int (\frac{1}{x ( x - 4 )}) d x

\int \frac{2 x ^{2}}{1 - 2 x ^{3}} d x

t an g e n t f (x) = (12 x - 3)^{\frac{1}{2}}, at x = 1