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Turunan dari $$$\sin{\left(5 \theta \right)}$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)$$$.
Solusi
Fungsi $$$\sin{\left(5 \theta \right)}$$$ merupakan komposisi $$$f{\left(g{\left(\theta \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ dan $$$g{\left(\theta \right)} = 5 \theta$$$.
Terapkan aturan rantai $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$:
$${\color{red}\left(\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{d\theta} \left(5 \theta\right)\right)}$$Turunan fungsi sinus adalah $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{d\theta} \left(5 \theta\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{d\theta} \left(5 \theta\right)$$Kembalikan ke variabel semula:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right) = \cos{\left({\color{red}\left(5 \theta\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right)$$Terapkan aturan kelipatan konstanta $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$ dengan $$$c = 5$$$ dan $$$f{\left(\theta \right)} = \theta$$$:
$$\cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(5 \theta\right)\right)} = \cos{\left(5 \theta \right)} {\color{red}\left(5 \frac{d}{d\theta} \left(\theta\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{d\theta} \left(\theta^{n}\right) = n \theta^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{d\theta} \left(\theta\right) = 1$$$:
$$5 \cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(\theta\right)\right)} = 5 \cos{\left(5 \theta \right)} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$.
Jawaban
$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$A