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$$$y^{2} = 10 x$$$ 對 $$$x$$$ 的隱式導數
您的輸入
求$$$\frac{d}{dx} \left(y^{2} = 10 x\right)$$$。
解答
分別對等式兩邊求導（將 $$$y$$$ 視為 $$$x$$$ 的函數）：$$$\frac{d}{dx} \left(y^{2}{\left(x \right)}\right) = \frac{d}{dx} \left(10 x\right)$$$。
對等式左邊求導數。
函數 $$$y^{2}{\left(x \right)}$$$ 是兩個函數 $$$f{\left(u \right)} = u^{2}$$$ 與 $$$g{\left(x \right)} = y{\left(x \right)}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(y^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(y{\left(x \right)}\right)\right)}$$套用冪次法則 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$，取 $$$n = 2$$$：
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(y{\left(x \right)}\right)$$返回原變數：
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) = 2 {\color{red}\left(y{\left(x \right)}\right)} \frac{d}{dx} \left(y{\left(x \right)}\right)$$因此，$$$\frac{d}{dx} \left(y^{2}{\left(x \right)}\right) = 2 y{\left(x \right)} \frac{d}{dx} \left(y{\left(x \right)}\right)$$$。
對等式右邊求導。
套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = 10$$$ 與 $$$f{\left(x \right)} = x$$$：
$${\color{red}\left(\frac{d}{dx} \left(10 x\right)\right)} = {\color{red}\left(10 \frac{d}{dx} \left(x\right)\right)}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$10 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 10 {\color{red}\left(1\right)}$$因此，$$$\frac{d}{dx} \left(10 x\right) = 10$$$。
因此，我們得到以下關於導數的線性方程：$$$2 y \frac{dy}{dx} = 10$$$。
解得 $$$\frac{dy}{dx} = \frac{5}{y}$$$。
答案
$$$\frac{dy}{dx} = \frac{5}{y}$$$A