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