derivative of (x+cos(x)^{cot(x)})

解答

\frac{d}{d x} ((x + cos (x))^{c o t (x)})

(- cos (x) + cot (x) - csc^{2} (x) ln (x + cos (x)) (x + cos (x))) (x + cos (x))^{c o t (x) - 1}

求解步骤

\frac{d}{d x} ((x + cos (x))^{c o t (x)})

使用指数法则:

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

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

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

使用连锁律:

e^{c o t (x) l n (x + c o s (x))} \frac{d}{d x} (cot (x) ln (x + cos (x)))

= e^{c o t (x) l n (x + c o s (x))} \frac{d}{d x} (cot (x) ln (x + cos (x)))

\frac{d}{d x} (cot (x) ln (x + cos (x))) = \frac{- cos ( x ) + cot ( x ) - csc ^{2} ( x ) ln ( x + cos ( x ) ) ( x + cos ( x ) )}{x + cos ( x )}

= e^{c o t (x) l n (x + c o s (x))} \frac{- cos ( x ) + cot ( x ) - csc ^{2} ( x ) ln ( x + cos ( x ) ) ( x + cos ( x ) )}{x + cos ( x )}

化简

e^{c o t (x) l n (x + c o s (x))} \frac{- cos ( x ) + cot ( x ) - csc ^{2} ( x ) ln ( x + cos ( x ) ) ( x + cos ( x ) )}{x + cos ( x )} : (- cos (x) + cot (x) - csc^{2} (x) ln (x + cos (x)) (x + cos (x))) (x + cos (x))^{c o t (x) - 1}

= (- cos (x) + cot (x) - csc^{2} (x) ln (x + cos (x)) (x + cos (x))) (x + cos (x))^{c o t (x) - 1}

y^{^{''}} - 2 y^{^{'}} + y = t e^{t} + 4, y (0) = 1, y^{^{'}} (0) = 1

d er i v a t i v e f (x) = x^{4} (3 - x)^{3}

d er i v a t i v e θ cos (θ) + sin (θ)

\int_{6}^{8} \frac{98}{( x - 6 ) ^{3}} d x

x \to 2 lim (- 1.999 + 2)