derivative 1/x sin(x)

解答

导数

\frac{1}{x} sin (x)

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

求解步骤

\frac{d}{d x} (\frac{1}{x} sin (x))

使用微分乘法定则:

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

f = \frac{1}{x}, g = sin (x)

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

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

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

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

化简

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

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

\frac{d}{d x} (\frac{sin ( 12 x )}{e ^{7 x}})

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