Turunan dari $$$x \left(x - 1\right)$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dx} \left(x \left(x - 1\right)\right)$$$.
Solusi
Terapkan aturan hasil kali $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ pada $$$f{\left(x \right)} = x$$$ dan $$$g{\left(x \right)} = x - 1$$$:
$${\color{red}\left(\frac{d}{dx} \left(x \left(x - 1\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \left(x - 1\right) + x \frac{d}{dx} \left(x - 1\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \frac{d}{dx} \left(x - 1\right) + \left(x - 1\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(x - 1\right) + \left(x - 1\right) {\color{red}\left(1\right)}$$Turunan dari jumlah/selisih adalah jumlah/selisih dari turunan:
$$x {\color{red}\left(\frac{d}{dx} \left(x - 1\right)\right)} + x - 1 = x {\color{red}\left(\frac{d}{dx} \left(x\right) - \frac{d}{dx} \left(1\right)\right)} + x - 1$$Turunan dari suatu konstanta adalah $$$0$$$:
$$x \left(- {\color{red}\left(\frac{d}{dx} \left(1\right)\right)} + \frac{d}{dx} \left(x\right)\right) + x - 1 = x \left(- {\color{red}\left(0\right)} + \frac{d}{dx} \left(x\right)\right) + x - 1$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + x - 1 = x {\color{red}\left(1\right)} + x - 1$$Dengan demikian, $$$\frac{d}{dx} \left(x \left(x - 1\right)\right) = 2 x - 1$$$.
Jawaban
$$$\frac{d}{dx} \left(x \left(x - 1\right)\right) = 2 x - 1$$$A