derivative of e^{1/x}sqrt(x^2-1)

解答

\frac{d}{d x} (e^{\frac{1}{x}} x^{2} - 1)

- \frac{e ^{\frac{1}{x}} x ^{2} - 1}{x ^{2}} + \frac{e ^{\frac{1}{x}} x}{( x ^{2} - 1 ) ^{\frac{1}{2}}}

求解步骤

\frac{d}{d x} (e^{\frac{1}{x}} x^{2} - 1)

使用微分乘法定则:

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

f = e^{\frac{1}{x}}, g = x^{2} - 1

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

\frac{d}{d x} (e^{\frac{1}{x}}) = - \frac{e ^{\frac{1}{x}}}{x ^{2}}

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

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

化简

(- \frac{e ^{\frac{1}{x}}}{x ^{2}}) x^{2} - 1 + \frac{x}{( x ^{2} - 1 ) ^{\frac{1}{2}}} e^{\frac{1}{x}} : - \frac{e ^{\frac{1}{x}} x ^{2} - 1}{x ^{2}} + \frac{e ^{\frac{1}{x}} x}{( x ^{2} - 1 ) ^{\frac{1}{2}}}

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

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