derivative of sin(xln(x^4+1/x))

解答

\frac{d}{d 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 )}

求解步骤

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

使用微分乘法定则:

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

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

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

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

\frac{d}{d 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)

分式相乘:

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 )}

\frac{d}{d x} (arcsin (sin (11 x)))

\int \frac{1}{t ^{2} - 2 t + 10} d t

\frac{\partial}{\partial x} (\frac{2 x}{9 - tan ( x )})

\int \frac{x ^{11}}{2 x ^{6} + 2} d x

y^{^{''}} + y^{^{'}} - 6 y = 2 x