derivative f(x)=e^xsqrt(1+x^2)

解答

导数

f (x) = e^{x} 1 + x^{2}

e^{x} 1 + x^{2} + \frac{e ^{x} x}{1 + x ^{2}}

求解步骤

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

使用微分乘法定则:

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

f = e^{x}, g = 1 + x^{2}

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

\frac{d}{d x} (e^{x}) = e^{x}

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

= e^{x} 1 + x^{2} + \frac{x}{1 + x ^{2}} e^{x}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= e^{x} 1 + x^{2} + \frac{e ^{x} x}{1 + x ^{2}}

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