derivative of {r}(xcos^2({r}(x)x))

解答

\frac{d}{d x} (r (x) cos^{2} (r (x) x))

cos^{2} (x r (x)) \frac{d}{d x} (r (x)) - 2 cos (x r (x)) sin (x r (x)) (x \frac{d}{d x} (r (x)) + r (x)) r (x)

求解步骤

\frac{d}{d x} (r (x) cos^{2} (r (x) x))

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = r (x), g = cos^{2} (r (x) x)

= \frac{d}{d x} (r (x)) cos^{2} (r (x) x) + \frac{d}{d x} (cos^{2} (r (x) x)) r (x)

\frac{d}{d x} (cos^{2} (r (x) x)) = - 2 cos (x r (x)) sin (x r (x)) (x \frac{d}{d x} (r (x)) + r (x))

= \frac{d}{d x} (r (x)) cos^{2} (r (x) x) + (- 2 cos (x r (x)) sin (x r (x)) (x \frac{d}{d x} (r (x)) + r (x))) r (x)

化简

= cos^{2} (x r (x)) \frac{d}{d x} (r (x)) - 2 cos (x r (x)) sin (x r (x)) (x \frac{d}{d x} (r (x)) + r (x)) r (x)

t an g e n t f (x) = \frac{2}{x + 1}, at x = 3, y = 1

d er i v a t i v e x - 2 x

\int \frac{e ^{- x}}{x} d x

3 x y + y^{2} + (x^{2} + x y) y^{^{'}} = 0

t \frac{d x}{d t} - x = 2 t^{2}