(\partial)/(\partial x)(sin(x)ln(x^4+1/x))

解

\frac{\partial}{\partial x} (sin (x) ln (x^{4} + \frac{1}{x}))

cos (x) ln (x^{4} + \frac{1}{x}) + \frac{sin ( x ) ( 4 x ^{5} - 1 )}{x ( x ^{5} + 1 )}

解答ステップ

\frac{\partial}{\partial x} (sin (x) ln (x^{4} + \frac{1}{x}))

積の法則を適用:

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

f = sin (x), g = ln (x^{4} + \frac{1}{x})

= \frac{\partial}{\partial x} (sin (x)) ln (x^{4} + \frac{1}{x}) + \frac{\partial}{\partial x} (ln (x^{4} + \frac{1}{x})) sin (x)

\frac{\partial}{\partial x} (sin (x)) = cos (x)

\frac{\partial}{\partial x} (ln (x^{4} + \frac{1}{x})) = \frac{4 x ^{5} - 1}{x ( x ^{5} + 1 )}

= cos (x) ln (x^{4} + \frac{1}{x}) + \frac{4 x ^{5} - 1}{x ( x ^{5} + 1 )} sin (x)

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= cos (x) ln (x^{4} + \frac{1}{x}) + \frac{sin ( x ) ( 4 x ^{5} - 1 )}{x ( x ^{5} + 1 )}