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