4.求由下列方程所确定的隐函数的二阶导数 dfrac ({d)^2y}(d{x)^2}-|||-(1) =1+x(e)^y;-|||-(2) ^2(x)^2+(a)^2(y)^2=(a)^2(b)^2

(1) $y=1+x{e}^{y}$

步骤 1：求一阶导数
对等式两边同时对 $x$ 求导，得到
$$
\dfrac{dy}{dx} = e^y + x e^y \dfrac{dy}{dx}
$$
整理得到
$$
\dfrac{dy}{dx} = \dfrac{e^y}{1 - x e^y}
$$

步骤 2：求二阶导数
对一阶导数再次对 $x$ 求导，得到
$$
\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\left(\dfrac{e^y}{1 - x e^y}\right)
$$
利用商的求导法则，得到
$$
\dfrac{d^2y}{dx^2} = \dfrac{(1 - x e^y) \dfrac{d}{dx}(e^y) - e^y \dfrac{d}{dx}(1 - x e^y)}{(1 - x e^y)^2}
$$
代入 $\dfrac{dy}{dx} = \dfrac{e^y}{1 - x e^y}$，得到
$$
\dfrac{d^2y}{dx^2} = \dfrac{(1 - x e^y) e^y \dfrac{dy}{dx} - e^y (-e^y - x e^y \dfrac{dy}{dx})}{(1 - x e^y)^2}
$$
化简得到
$$
\dfrac{d^2y}{dx^2} = \dfrac{e^{2y} (3 - y)}{(1 - x e^y)^3}
$$
代入 $y = 1 + x e^y$，得到
$$
\dfrac{d^2y}{dx^2} = \dfrac{e^{2y} (3 - y)}{(2 - y)^3}
$$

(2) ${b}^{2}{x}^{2}+{a}^{2}{y}^{2}={a}^{2}{b}^{2}$

步骤 1：求一阶导数
对等式两边同时对 $x$ 求导，得到
$$
2b^2x + 2a^2y \dfrac{dy}{dx} = 0
$$
整理得到
$$
\dfrac{dy}{dx} = -\dfrac{b^2x}{a^2y}
$$

步骤 2：求二阶导数
对一阶导数再次对 $x$ 求导，得到
$$
\dfrac{d^2y}{dx^2} = \dfrac{d}{dx}\left(-\dfrac{b^2x}{a^2y}\right)
$$
利用商的求导法则，得到
$$
\dfrac{d^2y}{dx^2} = -\dfrac{a^2y \dfrac{d}{dx}(b^2x) - b^2x \dfrac{d}{dx}(a^2y)}{(a^2y)^2}
$$
代入 $\dfrac{dy}{dx} = -\dfrac{b^2x}{a^2y}$，得到
$$
\dfrac{d^2y}{dx^2} = -\dfrac{a^2y b^2 - b^2x a^2 \dfrac{dy}{dx}}{(a^2y)^2}
$$
化简得到
$$
\dfrac{d^2y}{dx^2} = -\dfrac{b^4}{a^2y^3}
$$