Tentukan $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right)$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right)$$$.
Solusi
Tentukan turunan pertama $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)$$$
Turunan dari jumlah/selisih adalah jumlah/selisih dari turunan:
$${\color{red}\left(\frac{d}{dx} \left(x^{3} - 3 x^{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) - \frac{d}{dx} \left(3 x^{2}\right)\right)}$$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 $$$c = 3$$$ dan $$$f{\left(x \right)} = x^{2}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} + \frac{d}{dx} \left(x^{3}\right) = - {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(x^{3}\right)$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 2$$$:
$$- 3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(x^{3}\right) = - 3 {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(x^{3}\right)$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 3$$$:
$$- 6 x + {\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} = - 6 x + {\color{red}\left(3 x^{2}\right)}$$Sederhanakan:
$$3 x^{2} - 6 x = 3 x \left(x - 2\right)$$Dengan demikian, $$$\frac{d}{dx} \left(x^{3} - 3 x^{2}\right) = 3 x \left(x - 2\right)$$$.
Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(x^{3} - 3 x^{2}\right) = \frac{d}{dx} \left(3 x \left(x - 2\right)\right)$$$
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 $$$c = 3$$$ dan $$$f{\left(x \right)} = x \left(x - 2\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(3 x \left(x - 2\right)\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(x \left(x - 2\right)\right)\right)}$$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 $$$g{\left(x \right)} = x - 2$$$:
$$3 {\color{red}\left(\frac{d}{dx} \left(x \left(x - 2\right)\right)\right)} = 3 {\color{red}\left(\frac{d}{dx} \left(x\right) \left(x - 2\right) + x \frac{d}{dx} \left(x - 2\right)\right)}$$Turunan dari jumlah/selisih adalah jumlah/selisih dari turunan:
$$3 x {\color{red}\left(\frac{d}{dx} \left(x - 2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right) = 3 x {\color{red}\left(\frac{d}{dx} \left(x\right) - \frac{d}{dx} \left(2\right)\right)} + 3 \left(x - 2\right) \frac{d}{dx} \left(x\right)$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$3 x \left({\color{red}\left(\frac{d}{dx} \left(x\right)\right)} - \frac{d}{dx} \left(2\right)\right) + 3 \left(x - 2\right) {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 3 x \left({\color{red}\left(1\right)} - \frac{d}{dx} \left(2\right)\right) + 3 \left(x - 2\right) {\color{red}\left(1\right)}$$Turunan dari suatu konstanta adalah $$$0$$$:
$$3 x \left(1 - {\color{red}\left(\frac{d}{dx} \left(2\right)\right)}\right) + 3 x - 6 = 3 x \left(1 - {\color{red}\left(0\right)}\right) + 3 x - 6$$Dengan demikian, $$$\frac{d}{dx} \left(3 x \left(x - 2\right)\right) = 6 x - 6$$$.
Selanjutnya, $$$\frac{d^{3}}{dx^{3}} \left(x^{3} - 3 x^{2}\right) = \frac{d}{dx} \left(6 x - 6\right)$$$
Turunan dari jumlah/selisih adalah jumlah/selisih dari turunan:
$${\color{red}\left(\frac{d}{dx} \left(6 x - 6\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(6 x\right) - \frac{d}{dx} \left(6\right)\right)}$$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 $$$c = 6$$$ dan $$$f{\left(x \right)} = x$$$:
$${\color{red}\left(\frac{d}{dx} \left(6 x\right)\right)} - \frac{d}{dx} \left(6\right) = {\color{red}\left(6 \frac{d}{dx} \left(x\right)\right)} - \frac{d}{dx} \left(6\right)$$Turunan dari suatu konstanta adalah $$$0$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(6\right)\right)} + 6 \frac{d}{dx} \left(x\right) = - {\color{red}\left(0\right)} + 6 \frac{d}{dx} \left(x\right)$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$6 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 6 {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(6 x - 6\right) = 6$$$.
Selanjutnya, $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right) = \frac{d}{dx} \left(6\right)$$$
Turunan dari suatu konstanta adalah $$$0$$$:
$${\color{red}\left(\frac{d}{dx} \left(6\right)\right)} = {\color{red}\left(0\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(6\right) = 0$$$.
Oleh karena itu, $$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right) = 0$$$.
Jawaban
$$$\frac{d^{4}}{dx^{4}} \left(x^{3} - 3 x^{2}\right) = 0$$$A