derivative of (sin(0.5sqrt(x))/x)

解答

\frac{d}{d x} (\frac{sin ( 0.5 x )}{x})

\frac{0.25 x cos ( 0.5 x ) - sin ( 0.5 x )}{x ^{2}}

求解步骤

\frac{d}{d x} (\frac{sin ( 0.5 x )}{x})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( sin ( 0.5 x ) ) x - \frac{d x}{d x} sin ( 0.5 x )}{x ^{2}}

\frac{d}{d x} (sin (0.5 x)) = \frac{0.25 cos ( 0.5 x )}{x}

\frac{d x}{d x} = 1

= \frac{\frac{0.25 c o s ( 0.5 x )}{x} x - 1 \cdot sin ( 0.5 x )}{x ^{2}}

化简

\frac{\frac{0.25 c o s ( 0.5 x )}{x} x - 1 \cdot sin ( 0.5 x )}{x ^{2}} : \frac{0.25 x cos ( 0.5 x ) - sin ( 0.5 x )}{x ^{2}}

= \frac{0.25 x cos ( 0.5 x ) - sin ( 0.5 x )}{x ^{2}}

\int - \frac{2000 x}{( 1 + x ^{2} )} d x

t an g e n t x^{2}, at x = 2

\frac{\partial}{\partial x} (3 x y^{2} + 20 x^{2} y^{3})

y^{^{''}} + 2 y^{^{'}} + 2 y = 0, y (\frac{π}{4}) = 2, y^{^{'}} (\frac{π}{4}) = - 2

\int \frac{3 x ^{2} + 17 x + 32}{x ^{3} + 8 x ^{2} + 16 x} d x