derivative of x*cos(((x)/((x-2))))

解答

\frac{d}{d x} (x \cdot cos (\frac{( x )}{( x - 2 )}))

cos (\frac{x}{x - 2}) + \frac{2 x sin ( \frac{x}{x - 2} )}{( x - 2 ) ^{2}}

求解步骤

\frac{d}{d x} (x cos (\frac{( x )}{( x - 2 )}))

使用微分乘法定则:

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

f = x, g = cos (\frac{x}{x - 2})

= \frac{d x}{d x} cos (\frac{x}{x - 2}) + \frac{d}{d x} (cos (\frac{x}{x - 2})) x

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

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

= 1 \cdot cos (\frac{x}{x - 2}) + \frac{2 sin ( \frac{x}{x - 2} )}{( x - 2 ) ^{2}} x

化简

1 \cdot cos (\frac{x}{x - 2}) + \frac{2 sin ( \frac{x}{x - 2} )}{( x - 2 ) ^{2}} x : cos (\frac{x}{x - 2}) + \frac{2 x sin ( \frac{x}{x - 2} )}{( x - 2 ) ^{2}}

= cos (\frac{x}{x - 2}) + \frac{2 x sin ( \frac{x}{x - 2} )}{( x - 2 ) ^{2}}

x \to 0 lim (- e^{x})

x^{2} y^{^{'}} = (x + 1) y

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\int_{0}^{1} - 1 (x^{2} + 1) e^{- x} d x

\frac{\partial}{\partial x} (7 - 3 x^{2} - 3 y^{2})