derivative of 1/x y

解答

\frac{d}{d x} (\frac{1}{x} y)

- \frac{y}{x ^{2}} + \frac{\frac{d y}{d x}}{x}

求解步骤

\frac{d}{d x} (\frac{1}{x} y)

使用微分乘法定则:

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

f = \frac{1}{x}, g = y

= \frac{d}{d x} (\frac{1}{x}) y + \frac{d y}{d x} \frac{1}{x}

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

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

化简

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

= - \frac{y}{x ^{2}} + \frac{\frac{d y}{d x}}{x}

\frac{d}{d x} (2 e^{x} + \frac{4}{3 x})

x \to - 3 + lim (\frac{4 x + 3}{x + 3})

\frac{d y}{d x} = 2 x y^{\frac{1}{2}}

a re a 4 - x^{2}, 16 - 7 x

\int (9 - x) x d x