derivative of sin^n(xcos(n)x)

解答

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

cos (n) sin^{n} (x) (n x cot (x) + 1)

求解步骤

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

将常数提出:

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

= cos (n) \frac{d}{d x} (sin^{n} (x) x)

使用微分乘法定则:

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

f = sin^{n} (x), g = x

= cos (n) (\frac{d}{d x} (sin^{n} (x)) x + \frac{d x}{d x} sin^{n} (x))

\frac{d}{d x} (sin^{n} (x)) = n cos (x) sin^{n - 1} (x)

\frac{d x}{d x} = 1

= cos (n) (n cos (x) sin^{n - 1} (x) x + 1 \cdot sin^{n} (x))

化简

cos (n) (n cos (x) sin^{n - 1} (x) x + 1 \cdot sin^{n} (x)) : cos (n) sin^{n} (x) (n x cot (x) + 1)

= cos (n) sin^{n} (x) (n x cot (x) + 1)

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

ma c l a u r in (x + 3) (- 2 n (6 n - 1) (- 2 x)^{n - 1})

in v erse l a pl a ce \frac{s ^{2}}{s ^{2} + 6 s + 34}

\frac{d}{d x} (\frac{x - 3}{x + 1})

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