derivative 8/(sin(x)+cos(x))

解答

导数

\frac{8}{sin ( x ) + cos ( x )}

- \frac{8 ( cos ( x ) - sin ( x ) )}{1 + sin ( 2 x )}

求解步骤

\frac{d}{d x} (\frac{8}{sin ( x ) + cos ( x )})

将常数提出:

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

= 8 \frac{d}{d x} (\frac{1}{sin ( x ) + cos ( x )})

使用指数法则:

\frac{1}{a} = a^{- 1}

= 8 \frac{d}{d x} ((sin (x) + cos (x))^{- 1})

使用连锁律:

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

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

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

= 8 (- \frac{1}{( sin ( x ) + cos ( x ) ) ^{2}} (cos (x) - sin (x)))

化简

8 (- \frac{1}{( sin ( x ) + cos ( x ) ) ^{2}} (cos (x) - sin (x))) : - \frac{8 ( cos ( x ) - sin ( x ) )}{1 + sin ( 2 x )}

= - \frac{8 ( cos ( x ) - sin ( x ) )}{1 + sin ( 2 x )}

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