derivative of x^{-1/2}*cos(x)

解答

\frac{d}{d x} (x^{- \frac{1}{2}} \cdot cos (x))

- \frac{cos ( x )}{2 x ^{\frac{3}{2}}} - \frac{sin ( x )}{x ^{\frac{1}{2}}}

求解步骤

\frac{d}{d x} (x^{- \frac{1}{2}} cos (x))

使用微分乘法定则:

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

f = x^{- \frac{1}{2}}, g = cos (x)

= \frac{d}{d x} (x^{- \frac{1}{2}}) cos (x) + \frac{d}{d x} (cos (x)) x^{- \frac{1}{2}}

\frac{d}{d x} (x^{- \frac{1}{2}}) = - \frac{1}{2 x ^{\frac{3}{2}}}

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

= (- \frac{1}{2 x ^{\frac{3}{2}}}) cos (x) + (- sin (x)) x^{- \frac{1}{2}}

化简

(- \frac{1}{2 x ^{\frac{3}{2}}}) cos (x) + (- sin (x)) x^{- \frac{1}{2}} : - \frac{cos ( x )}{2 x ^{\frac{3}{2}}} - \frac{sin ( x )}{x ^{\frac{1}{2}}}

= - \frac{cos ( x )}{2 x ^{\frac{3}{2}}} - \frac{sin ( x )}{x ^{\frac{1}{2}}}

l a pl a ce t r an s f or m sin (\frac{2 t}{3})

\frac{d}{d x} (\frac{x}{( x - 7 ) ^{2}})

in v erse l a pl a ce \frac{6 s - 5}{s ^{2} + 7}

y^{^{'}} - \frac{2 y}{x} = x^{3} y^{- 3}

\frac{d}{d x} (- e^{- x} x)