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Turunan implisit dari $$$y^{2} = 10 x$$$ terhadap $$$x$$$
Masukan Anda
Temukan $$$\frac{d}{dx} \left(y^{2} = 10 x\right)$$$.
Solusi
Turunkan secara terpisah kedua ruas persamaan (anggap $$$y$$$ sebagai fungsi dari $$$x$$$): $$$\frac{d}{dx} \left(y^{2}{\left(x \right)}\right) = \frac{d}{dx} \left(10 x\right)$$$.
Turunkan ruas kiri dari persamaan.
Fungsi $$$y^{2}{\left(x \right)}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = u^{2}$$$ dan $$$g{\left(x \right)} = y{\left(x \right)}$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(y^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(y{\left(x \right)}\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ dengan $$$n = 2$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(y{\left(x \right)}\right)$$Kembalikan ke variabel semula:
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) = 2 {\color{red}\left(y{\left(x \right)}\right)} \frac{d}{dx} \left(y{\left(x \right)}\right)$$Dengan demikian, $$$\frac{d}{dx} \left(y^{2}{\left(x \right)}\right) = 2 y{\left(x \right)} \frac{d}{dx} \left(y{\left(x \right)}\right)$$$.
Turunkan ruas kanan persamaan.
Terapkan aturan kelipatan konstanta $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ dengan $$$c = 10$$$ dan $$$f{\left(x \right)} = x$$$:
$${\color{red}\left(\frac{d}{dx} \left(10 x\right)\right)} = {\color{red}\left(10 \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$10 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 10 {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(10 x\right) = 10$$$.
Dengan demikian, kita memperoleh persamaan linier berikut terhadap turunan: $$$2 y \frac{dy}{dx} = 10$$$.
Dengan menyelesaikannya, kita memperoleh bahwa $$$\frac{dy}{dx} = \frac{5}{y}$$$.
Jawaban
$$$\frac{dy}{dx} = \frac{5}{y}$$$A