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

解答

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

sin^{x} (x) (\frac{ln ( sin ( x ) )}{2 x} + x cot (x))

求解步骤

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

使用指数法则:

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

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

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

使用连锁律:

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

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

\frac{d}{d x} (x ln (sin (x))) = \frac{ln ( sin ( x ) )}{2 x} + x cot (x)

= e^{x l n (s i n (x))} (\frac{ln ( sin ( x ) )}{2 x} + x cot (x))

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

= sin^{x} (x) (\frac{ln ( sin ( x ) )}{2 x} + x cot (x))

t an g e n t f (x) = 5 x - 9 x^{2}, at x = 1

\frac{d}{d x} (- 8 cos (x))

d er i v a t i v e e^{x} x + 2 e^{x}

\int \frac{8 + x}{64 - x ^{2}} d x

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