derivative of (sqrt(x^2+1)/x)

解答

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

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

求解步骤

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

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

\frac{d x}{d x} = 1

= \frac{\frac{x}{x ^{2} + 1} x - 1 \cdot x ^{2} + 1}{x ^{2}}

化简

\frac{\frac{x}{x ^{2} + 1} x - 1 \cdot x ^{2} + 1}{x ^{2}} : - \frac{1}{x ^{2} x ^{2} + 1}

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

\frac{d}{d x} (\frac{1}{x ^{3} + 2})

x \to 2 lim (x^{3} - 2 x^{2} + x - 3)

d er i v a t i v e (2 - 3 x^{2})^{4} (x^{7} + 3)^{3}

d er i v a t i v e f (x) = \frac{x}{4}

\frac{d}{d x} (2 + 5 x)