derivative of x^{1/4}cos(sqrt(7)ln(x))

解

\frac{d}{d x} (x^{\frac{1}{4}} cos (7 ln (x)))

\frac{cos ( 7 ln ( x ) )}{4 x ^{\frac{3}{4}}} - \frac{7 ^{\frac{1}{2}} sin ( 7 ln ( x ) )}{x ^{\frac{3}{4}}}

解答ステップ

\frac{d}{d x} (x^{\frac{1}{4}} cos (7 ln (x)))

積の法則を適用:

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

f = x^{\frac{1}{4}}, g = cos (7 ln (x))

= \frac{d}{d x} (x^{\frac{1}{4}}) cos (7 ln (x)) + \frac{d}{d x} (cos (7 ln (x))) x^{\frac{1}{4}}

\frac{d}{d x} (x^{\frac{1}{4}}) = \frac{1}{4 x ^{\frac{3}{4}}}

\frac{d}{d x} (cos (7 ln (x))) = - \frac{7 sin ( 7 ln ( x ) )}{x}

= \frac{1}{4 x ^{\frac{3}{4}}} cos (7 ln (x)) + (- \frac{7 sin ( 7 ln ( x ) )}{x}) x^{\frac{1}{4}}

簡素化

\frac{1}{4 x ^{\frac{3}{4}}} cos (7 ln (x)) + (- \frac{7 sin ( 7 ln ( x ) )}{x}) x^{\frac{1}{4}} : \frac{cos ( 7 ln ( x ) )}{4 x ^{\frac{3}{4}}} - \frac{7 ^{\frac{1}{2}} sin ( 7 ln ( x ) )}{x ^{\frac{3}{4}}}

= \frac{cos ( 7 ln ( x ) )}{4 x ^{\frac{3}{4}}} - \frac{7 ^{\frac{1}{2}} sin ( 7 ln ( x ) )}{x ^{\frac{3}{4}}}