derivative of (y^2-3x^2/(y^4))

解答

\frac{d}{d x} (\frac{y ^{2} - 3 x ^{2}}{y ^{4}})

\frac{2 ( 6 x ^{2} \frac{d y}{d x} - 3 x y - \frac{d y}{d x} y ^{2} )}{y ^{5}}

求解步骤

\frac{d}{d x} (\frac{y ^{2} - 3 x ^{2}}{y ^{4}})

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} ( y ^{2} - 3 x ^{2} ) y ^{4} - \frac{d}{d x} ( y ^{4} ) ( y ^{2} - 3 x ^{2} )}{( y ^{4} ) ^{2}}

\frac{d}{d x} (y^{2} - 3 x^{2}) = 2 y \frac{d y}{d x} - 6 x

\frac{d}{d x} (y^{4}) = 4 y^{3} \frac{d y}{d x}

= \frac{( 2 y \frac{d y}{d x} - 6 x ) y ^{4} - 4 y ^{3} \frac{d y}{d x} ( y ^{2} - 3 x ^{2} )}{( y ^{4} ) ^{2}}

化简

\frac{( 2 y \frac{d y}{d x} - 6 x ) y ^{4} - 4 y ^{3} \frac{d y}{d x} ( y ^{2} - 3 x ^{2} )}{( y ^{4} ) ^{2}} : \frac{2 ( 6 x ^{2} \frac{d y}{d x} - 3 x y - \frac{d y}{d x} y ^{2} )}{y ^{5}}

= \frac{2 ( 6 x ^{2} \frac{d y}{d x} - 3 x y - \frac{d y}{d x} y ^{2} )}{y ^{5}}

\frac{\partial}{\partial x} (+ 6 - 5 + 6 x)

n = 0 \sum \infty \frac{2}{7 ^{n}}

x \to 2 - lim (arcsin (x - 1))

x \to π lim (\frac{1}{x - π})

\int \frac{1}{5 x ^{2}} d x