derivative of 3csc(xcos^3(x))

解答

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

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

求解步骤

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

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

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

使用微分乘法定则:

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

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

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

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

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

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

化简

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

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

t an g e n t y = \frac{1}{1 + x ^{2}}, (- 1, \frac{1}{2})

in v erse l a pl a ce \frac{2}{sin ^{2} ( + 2 )}

t a y l or f (x) = x

x \to 0 lim (cos (π x))

\int e^{- 2 x} x^{2} d x