derivative f(x)=(x^2)/2*cos^3(1/x)

解

導関数

f (x) = \frac{x ^{2}}{2} \cdot cos^{3} (\frac{1}{x})

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

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

f = x^{2}, g = cos^{3} (\frac{1}{x})

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

\frac{d}{d x} (x^{2}) = 2 x

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

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

簡素化

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

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