derivative of (sin(xsec(x))/(1+xtan(x)))

解答

\frac{d}{d x} (\frac{sin ( x ) sec ( x )}{1 + x tan ( x )})

\frac{1}{( 1 + x tan ( x ) ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{sin ( x ) sec ( x )}{1 + x tan ( x )})

使用微分除法定则:

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

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

\frac{d}{d x} (sin (x) sec (x)) = sec^{2} (x)

\frac{d}{d x} (1 + x tan (x)) = x sec^{2} (x) + tan (x)

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

化简

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

= \frac{1}{( 1 + x tan ( x ) ) ^{2}}