derivative of ((x-y)/(y_{2)(x)})

解答

\frac{d}{d x} (\frac{( x - y )}{y _{2} ( x ) ( x )})

\frac{- x ^{2} \frac{d}{d x} ( y _{2} ( x ) ) - x y _{2} ( x ) \frac{d y}{d x} + x y \frac{d}{d x} ( y _{2} ( x ) ) + y y _{2} ( x )}{x ^{2} y _{2} ( x ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{( x - y )}{y _{2} ( x ) ( 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} ( x - y ) y _{2} ( x ) x - \frac{d}{d x} ( y _{2} ( x ) x ) ( x - y )}{( y _{2} ( x ) x ) ^{2}}

\frac{d}{d x} (x - y) = 1 - \frac{d y}{d x}

\frac{d}{d x} (y_{2} (x) x) = x \frac{d}{d x} (y_{2} (x)) + y_{2} (x)

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

化简

\frac{( 1 - \frac{d y}{d x} ) y _{2} ( x ) x - ( x \frac{d}{d x} ( y _{2} ( x ) ) + y _{2} ( x ) ) ( x - y )}{( y _{2} ( x ) x ) ^{2}} : \frac{- x ^{2} \frac{d}{d x} ( y _{2} ( x ) ) - x y _{2} ( x ) \frac{d y}{d x} + x y \frac{d}{d x} ( y _{2} ( x ) ) + y y _{2} ( x )}{x ^{2} y _{2} ( x ) ^{2}}

= \frac{- x ^{2} \frac{d}{d x} ( y _{2} ( x ) ) - x y _{2} ( x ) \frac{d y}{d x} + x y \frac{d}{d x} ( y _{2} ( x ) ) + y y _{2} ( x )}{x ^{2} y _{2} ( x ) ^{2}}

d er i v a t i v e 2 cos^{2} (\frac{t}{2})

d er i v a t i v e 2 x - (\frac{250}{x ^{2}})

d er i v a t i v e sin (x^{6}) + sin^{6} (x)

d er i v a t i v e 3 \frac{x ^{2}}{x} - \frac{2}{x}

d er i v a t i v e lo g_{10} (x) x + 1