derivative of (sin(7x)^{ln(x)})

解答

\frac{d}{d x} ((sin (7 x))^{l n (x)})

x^{l n (s i n (7 x))} (\frac{ln ( sin ( 7 x ) )}{x} + 7 cot (7 x) ln (x))

求解步骤

\frac{d}{d x} ((sin (7 x))^{l n (x)})

使用指数法则:

a^{b} = e^{b l n (a)}

(sin (7 x))^{l n (x)} = e^{l n (x) l n (s i n (7 x))}

= \frac{d}{d x} (e^{l n (x) l n (s i n (7 x))})

使用连锁律:

e^{l n (x) l n (s i n (7 x))} \frac{d}{d x} (ln (x) ln (sin (7 x)))

= e^{l n (x) l n (s i n (7 x))} \frac{d}{d x} (ln (x) ln (sin (7 x)))

\frac{d}{d x} (ln (x) ln (sin (7 x))) = \frac{ln ( sin ( 7 x ) )}{x} + 7 cot (7 x) ln (x)

= e^{l n (x) l n (s i n (7 x))} (\frac{ln ( sin ( 7 x ) )}{x} + 7 cot (7 x) ln (x))

e^{l n (x) l n (s i n (7 x))} = x^{l n (s i n (7 x))}

= x^{l n (s i n (7 x))} (\frac{ln ( sin ( 7 x ) )}{x} + 7 cot (7 x) ln (x))

\frac{d}{d x} (1 - e^{- \frac{x}{3}})

(\frac{5}{x ^{5}})^{^{'}}

a re a 2 y = 3 x, y = 3, 2 y + 4 x = 7

\int_{0}^{\frac{π}{4}} 1 + tan (x) d x

\frac{d}{d x} (\frac{6 x}{x ^{2} - 9})