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