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

解答

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

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

求解步骤

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

使用指数法则:

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

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

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

使用连锁律:

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

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

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

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

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

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

\frac{\partial}{\partial x} (x^{2} y + 5 y^{3})

\frac{d x}{d t} = 3 t^{2} (x^{2} + 4)

\frac{\partial}{\partial x} (9 \frac{x yz}{x + y + z})

p a r i t y x^{c o s (x)}

\frac{d}{d x} (5^{- 2 x})