derivative of (x^9e^x/(x^9+e^x))

解答

\frac{d}{d x} (\frac{x ^{9} e ^{x}}{x ^{9} + e ^{x}})

\frac{e ^{x} x ^{18} + 9 e ^{2 x} x ^{8}}{( x ^{9} + e ^{x} ) ^{2}}

求解步骤

\frac{d}{d x} (\frac{x ^{9} e ^{x}}{x ^{9} + e ^{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 ^{9} e ^{x} ) ( x ^{9} + e ^{x} ) - \frac{d}{d x} ( x ^{9} + e ^{x} ) x ^{9} e ^{x}}{( x ^{9} + e ^{x} ) ^{2}}

\frac{d}{d x} (x^{9} e^{x}) = 9 x^{8} e^{x} + e^{x} x^{9}

\frac{d}{d x} (x^{9} + e^{x}) = 9 x^{8} + e^{x}

= \frac{( 9 x ^{8} e ^{x} + e ^{x} x ^{9} ) ( x ^{9} + e ^{x} ) - ( 9 x ^{8} + e ^{x} ) x ^{9} e ^{x}}{( x ^{9} + e ^{x} ) ^{2}}

乘开

(9 x^{8} e^{x} + e^{x} x^{9}) (x^{9} + e^{x}) - (9 x^{8} + e^{x}) x^{9} e^{x} : e^{x} x^{18} + 9 e^{2 x} x^{8}

= \frac{e ^{x} x ^{18} + 9 e ^{2 x} x ^{8}}{( x ^{9} + e ^{x} ) ^{2}}

ma c l a u r in cos (θ)

im pl i c i t \frac{d y}{d x}, (3 x y + 7)^{2} = 6 y

\int \frac{3}{8} x^{2} d x

a re a y = x, y = x - 2, y = 0

t an g e n t f (x) = 3 x^{- 2}, at x = 4