derivative f(x)=(6cos(x))/(7-9cos(x))

解答

导数

f (x) = \frac{6 cos ( x )}{7 - 9 cos ( x )}

- \frac{42 sin ( x )}{( 7 - 9 cos ( x ) ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{6 cos ( x )}{7 - 9 cos ( x )})

将常数提出:

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

= 6 \frac{d}{d x} (\frac{cos ( x )}{7 - 9 cos ( x )})

使用微分除法定则:

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

= 6 \cdot \frac{\frac{d}{d x} ( cos ( x ) ) ( 7 - 9 cos ( x ) ) - \frac{d}{d x} ( 7 - 9 cos ( x ) ) cos ( x )}{( 7 - 9 cos ( x ) ) ^{2}}

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

\frac{d}{d x} (7 - 9 cos (x)) = 9 sin (x)

= 6 \cdot \frac{( - sin ( x ) ) ( 7 - 9 cos ( x ) ) - 9 sin ( x ) cos ( x )}{( 7 - 9 cos ( x ) ) ^{2}}

化简

6 \cdot \frac{( - sin ( x ) ) ( 7 - 9 cos ( x ) ) - 9 sin ( x ) cos ( x )}{( 7 - 9 cos ( x ) ) ^{2}} : - \frac{42 sin ( x )}{( 7 - 9 cos ( x ) ) ^{2}}

= - \frac{42 sin ( x )}{( 7 - 9 cos ( x ) ) ^{2}}

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