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Turunan dari $$$\sin{\left(\frac{\pi x}{3} \right)}$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dx} \left(\sin{\left(\frac{\pi x}{3} \right)}\right)$$$.
Solusi
Fungsi $$$\sin{\left(\frac{\pi x}{3} \right)}$$$ merupakan komposisi $$$f{\left(g{\left(x \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ dan $$$g{\left(x \right)} = \frac{\pi x}{3}$$$.
Terapkan aturan rantai $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(\frac{\pi x}{3} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(\frac{\pi x}{3}\right)\right)}$$Turunan fungsi sinus adalah $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{\pi x}{3}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{\pi x}{3}\right)$$Kembalikan ke variabel semula:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{\pi x}{3}\right) = \cos{\left({\color{red}\left(\frac{\pi x}{3}\right)} \right)} \frac{d}{dx} \left(\frac{\pi x}{3}\right)$$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 $$$c = \frac{\pi}{3}$$$ dan $$$f{\left(x \right)} = x$$$:
$$\cos{\left(\frac{\pi x}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{\pi x}{3}\right)\right)} = \cos{\left(\frac{\pi x}{3} \right)} {\color{red}\left(\frac{\pi}{3} \frac{d}{dx} \left(x\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\frac{\pi \cos{\left(\frac{\pi x}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{3} = \frac{\pi \cos{\left(\frac{\pi x}{3} \right)} {\color{red}\left(1\right)}}{3}$$Dengan demikian, $$$\frac{d}{dx} \left(\sin{\left(\frac{\pi x}{3} \right)}\right) = \frac{\pi \cos{\left(\frac{\pi x}{3} \right)}}{3}$$$.
Jawaban
$$$\frac{d}{dx} \left(\sin{\left(\frac{\pi x}{3} \right)}\right) = \frac{\pi \cos{\left(\frac{\pi x}{3} \right)}}{3}$$$A