derivative of 5/3 x^2y^{-2}

解答

\frac{d}{d x} (\frac{5}{3} x^{2} y^{- 2})

\frac{5}{3} (\frac{2 x}{y ^{2}} - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}})

求解步骤

\frac{d}{d x} (\frac{5}{3} x^{2} y^{- 2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{5}{3} \frac{d}{d x} (x^{2} y^{- 2})

使用微分乘法定则:

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

f = x^{2}, g = y^{- 2}

= \frac{5}{3} (\frac{d}{d x} (x^{2}) y^{- 2} + \frac{d}{d x} (y^{- 2}) x^{2})

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

\frac{d}{d x} (y^{- 2}) = - \frac{2 \frac{d y}{d x}}{y ^{3}}

= \frac{5}{3} (2 x y^{- 2} + (- \frac{2 \frac{d y}{d x}}{y ^{3}}) x^{2})

化简

\frac{5}{3} (2 x y^{- 2} + (- \frac{2 \frac{d y}{d x}}{y ^{3}}) x^{2}) : \frac{5}{3} (\frac{2 x}{y ^{2}} - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}})

= \frac{5}{3} (\frac{2 x}{y ^{2}} - \frac{2 x ^{2} \frac{d y}{d x}}{y ^{3}})

\int \frac{2}{x ^{2} + 9} d x

n = 0 \sum \infty \frac{n}{1 + n ^{2}}

t an g e n t f (x) = \frac{2}{x ^{2}}, (7, \frac{2}{49})

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

\frac{d}{d y} (\frac{y}{y + \frac{b}{y}})