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Turunan kedua dari $$$\cos^{2}{\left(x \right)}$$$
Kalkulator terkait: Kalkulator Turunan, Kalkulator Diferensiasi Logaritmik
Masukan Anda
Temukan $$$\frac{d^{2}}{dx^{2}} \left(\cos^{2}{\left(x \right)}\right)$$$.
Solusi
Tentukan turunan pertama $$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)$$$
Fungsi $$$\cos^{2}{\left(x \right)}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = u^{2}$$$ dan $$$g{\left(x \right)} = \cos{\left(x \right)}$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ dengan $$$n = 2$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$Kembalikan ke variabel semula:
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = 2 {\color{red}\left(\cos{\left(x \right)}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$Turunan fungsi kosinus adalah $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$2 \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = 2 \cos{\left(x \right)} {\color{red}\left(- \sin{\left(x \right)}\right)}$$Sederhanakan:
$$- 2 \sin{\left(x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)}$$Dengan demikian, $$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right) = - \sin{\left(2 x \right)}$$$.
Selanjutnya, $$$\frac{d^{2}}{dx^{2}} \left(\cos^{2}{\left(x \right)}\right) = \frac{d}{dx} \left(- \sin{\left(2 x \right)}\right)$$$
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 $$$c = -1$$$ dan $$$f{\left(x \right)} = \sin{\left(2 x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \sin{\left(2 x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)}$$Fungsi $$$\sin{\left(2 x \right)}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ dan $$$g{\left(x \right)} = 2 x$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\sin{\left(2 x \right)}\right)\right)} = - {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(2 x\right)\right)}$$Turunan fungsi sinus adalah $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$$- {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(2 x\right) = - {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(2 x\right)$$Kembalikan ke variabel semula:
$$- \cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(2 x\right) = - \cos{\left({\color{red}\left(2 x\right)} \right)} \frac{d}{dx} \left(2 x\right)$$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 $$$c = 2$$$ dan $$$f{\left(x \right)} = x$$$:
$$- \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} = - \cos{\left(2 x \right)} {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- 2 \cos{\left(2 x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 2 \cos{\left(2 x \right)} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dx} \left(- \sin{\left(2 x \right)}\right) = - 2 \cos{\left(2 x \right)}$$$.
Oleh karena itu, $$$\frac{d^{2}}{dx^{2}} \left(\cos^{2}{\left(x \right)}\right) = - 2 \cos{\left(2 x \right)}$$$.
Jawaban
$$$\frac{d^{2}}{dx^{2}} \left(\cos^{2}{\left(x \right)}\right) = - 2 \cos{\left(2 x \right)}$$$A