derivative f(x)=(sin(5x))/(3x)

解答

导数

f (x) = \frac{sin ( 5 x )}{3 x}

\frac{5 x cos ( 5 x ) - sin ( 5 x )}{3 x ^{2}}

求解步骤

\frac{d}{d x} (\frac{sin ( 5 x )}{3 x})

将常数提出:

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

= \frac{1}{3} \frac{d}{d x} (\frac{sin ( 5 x )}{x})

使用微分除法定则:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{1}{3} \cdot \frac{\frac{d}{d x} ( sin ( 5 x ) ) x - \frac{d x}{d x} sin ( 5 x )}{x ^{2}}

\frac{d}{d x} (sin (5 x)) = cos (5 x) \cdot 5

\frac{d x}{d x} = 1

= \frac{1}{3} \cdot \frac{cos ( 5 x ) \cdot 5 x - 1 \cdot sin ( 5 x )}{x ^{2}}

化简

\frac{1}{3} \cdot \frac{cos ( 5 x ) \cdot 5 x - 1 \cdot sin ( 5 x )}{x ^{2}} : \frac{5 x cos ( 5 x ) - sin ( 5 x )}{3 x ^{2}}

= \frac{5 x cos ( 5 x ) - sin ( 5 x )}{3 x ^{2}}

\int \frac{1}{1 - 49 x ^{2}} d x

\int \frac{x ^{2} - 25}{x - 5} d x

\frac{d}{d x} (7 e^{x} + \frac{2}{3 x})

\int_{0}^{π} sec^{2} (x) d x

d er i v a t i v e y = 8 arcsin (\frac{x}{4}) - \frac{x 16 - x ^{2}}{2}