derivative f(x)=csc(x)(x+cot(x))

解答

导数

f (x) = csc (x) (x + cot (x))

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

求解步骤

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

使用微分乘法定则:

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

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

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

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

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

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

化简

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

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

\frac{d}{d x} (4 x^{\frac{5}{4}} + 6 x^{\frac{3}{2}} + 9 x)

\int x^{2} - y d y

x \to 0 lim (\frac{x ^{2}}{2 x})

\int_{0}^{1} 2 π \cdot \frac{2}{3} \cdot x \cdot \frac{2}{3} d x

\frac{d}{d x} (ln (8 x - 5) \cdot 7^{5 x - 2})