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Turunan kedua dari $$$x \sin{\left(x \right)}$$$

Kalkulator akan menghitung turunan kedua dari $$$x \sin{\left(x \right)}$$$, beserta langkah-langkahnya.

Kalkulator terkait: Kalkulator Turunan, Kalkulator Diferensiasi Logaritmik

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Masukan Anda

Temukan $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right)$$$.

Solusi

Tentukan turunan pertama $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right)$$$

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 $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(x \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(x \right)} + x \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$

Turunan fungsi sinus adalah $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$$x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right) = x {\color{red}\left(\cos{\left(x \right)}\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right)$$

Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(1\right)}$$

Dengan demikian, $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right) = x \cos{\left(x \right)} + \sin{\left(x \right)}$$$.

Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = \frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$$$

Turunan dari jumlah/selisih adalah jumlah/selisih dari turunan:

$${\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right) + \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$

Turunan fungsi sinus adalah $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right) = {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right)$$

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 $$$g{\left(x \right)} = \cos{\left(x \right)}$$$:

$$\cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right)\right)} = \cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(x\right) \cos{\left(x \right)} + x \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$

Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \cos{\left(x \right)} = x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(1\right)} + \cos{\left(x \right)}$$

Turunan fungsi kosinus adalah $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:

$$x {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + 2 \cos{\left(x \right)} = x {\color{red}\left(- \sin{\left(x \right)}\right)} + 2 \cos{\left(x \right)}$$

Dengan demikian, $$$\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$.

Oleh karena itu, $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$.

Jawaban

$$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$A


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