derivative of cos(tan(x)e^{-(x/8)^2})

解答

\frac{d}{d x} (cos (tan (x)) e^{- (\frac{x}{8})^{2}})

- \frac{e ^{- \frac{x ^{2}}{64}} x cos ( tan ( x ) )}{32} - e^{- \frac{x ^{2}}{64}} sec^{2} (x) sin (tan (x))

求解步骤

\frac{d}{d x} (cos (tan (x)) e^{- (\frac{x}{8})^{2}})

化简

cos (tan (x)) e^{- (\frac{x}{8})^{2}} : e^{- \frac{x ^{2}}{64}} cos (tan (x))

= \frac{d}{d x} (e^{- \frac{x ^{2}}{64}} cos (tan (x)))

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{- \frac{x ^{2}}{64}}, g = cos (tan (x))

= \frac{d}{d x} (e^{- \frac{x ^{2}}{64}}) cos (tan (x)) + \frac{d}{d x} (cos (tan (x))) e^{- \frac{x ^{2}}{64}}

\frac{d}{d x} (e^{- \frac{x ^{2}}{64}}) = - \frac{e ^{- \frac{x ^{2}}{64}} x}{32}

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

= (- \frac{e ^{- \frac{x ^{2}}{64}} x}{32}) cos (tan (x)) + (- sin (tan (x)) sec^{2} (x)) e^{- \frac{x ^{2}}{64}}

化简

(- \frac{e ^{- \frac{x ^{2}}{64}} x}{32}) cos (tan (x)) + (- sin (tan (x)) sec^{2} (x)) e^{- \frac{x ^{2}}{64}} : - \frac{e ^{- \frac{x ^{2}}{64}} x cos ( tan ( x ) )}{32} - e^{- \frac{x ^{2}}{64}} sec^{2} (x) sin (tan (x))

= - \frac{e ^{- \frac{x ^{2}}{64}} x cos ( tan ( x ) )}{32} - e^{- \frac{x ^{2}}{64}} sec^{2} (x) sin (tan (x))

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