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

解答

\frac{d}{d x} ((cos (x))^{s i n (x)})

cos^{s i n (x)} (x) (cos (x) ln (cos (x)) - tan (x) sin (x))

求解步骤

\frac{d}{d x} ((cos (x))^{s i n (x)})

使用指数法则:

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

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

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

使用连锁律:

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

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

\frac{d}{d x} (sin (x) ln (cos (x))) = cos (x) ln (cos (x)) - tan (x) sin (x)

= e^{s i n (x) l n (c o s (x))} (cos (x) ln (cos (x)) - tan (x) sin (x))

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

= cos^{s i n (x)} (x) (cos (x) ln (cos (x)) - tan (x) sin (x))

a re a e^{x}, 14 e^{- x} + 5, x = 0

\int x (4 x^{2} + 8) d x

\int x 1 - x^{4} d x

t an g e n t f (x) = 2 x, at x = 8

\int_{0}^{1} 2 (x^{2} + 1) e^{- x} d x