implicit (dy)/(dx),(xy)^{1/2}=1+x^2y

解答

隐函数求导

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

\frac{d y}{d x} = \frac{4 x ^{\frac{3}{2}} y ^{\frac{3}{2}} - y}{x - 2 x ^{\frac{5}{2}} y ^{\frac{1}{2}}}

求解步骤

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

将

y

视为

y (x)

对两侧微分:

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

整理

\frac{d y}{d x} : \frac{d y}{d x} = \frac{4 x ^{\frac{3}{2}} y ^{\frac{3}{2}} - y}{x - 2 x ^{\frac{5}{2}} y ^{\frac{1}{2}}}

\frac{d y}{d x} = \frac{4 x ^{\frac{3}{2}} y ^{\frac{3}{2}} - y}{x - 2 x ^{\frac{5}{2}} y ^{\frac{1}{2}}}