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Turunan dari $$$e^{x} + \sin{\left(y z \right)}$$$ terhadap $$$z$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right)$$$.
Solusi
Turunan dari jumlah/selisih adalah jumlah/selisih dari turunan:
$${\color{red}\left(\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right)\right)} = {\color{red}\left(\frac{d}{dz} \left(e^{x}\right) + \frac{d}{dz} \left(\sin{\left(y z \right)}\right)\right)}$$Turunan dari suatu konstanta adalah $$$0$$$:
$${\color{red}\left(\frac{d}{dz} \left(e^{x}\right)\right)} + \frac{d}{dz} \left(\sin{\left(y z \right)}\right) = {\color{red}\left(0\right)} + \frac{d}{dz} \left(\sin{\left(y z \right)}\right)$$Fungsi $$$\sin{\left(y z \right)}$$$ merupakan komposisi $$$f{\left(g{\left(z \right)} \right)}$$$ dari dua fungsi $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ dan $$$g{\left(z \right)} = y z$$$.
Terapkan aturan rantai $$$\frac{d}{dz} \left(f{\left(g{\left(z \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dz} \left(g{\left(z \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dz} \left(\sin{\left(y z \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dz} \left(y z\right)\right)}$$Turunan fungsi sinus adalah $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dz} \left(y z\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dz} \left(y z\right)$$Kembalikan ke variabel semula:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dz} \left(y z\right) = \cos{\left({\color{red}\left(y z\right)} \right)} \frac{d}{dz} \left(y z\right)$$Terapkan aturan kelipatan konstanta $$$\frac{d}{dz} \left(c f{\left(z \right)}\right) = c \frac{d}{dz} \left(f{\left(z \right)}\right)$$$ dengan $$$c = y$$$ dan $$$f{\left(z \right)} = z$$$:
$$\cos{\left(y z \right)} {\color{red}\left(\frac{d}{dz} \left(y z\right)\right)} = \cos{\left(y z \right)} {\color{red}\left(y \frac{d}{dz} \left(z\right)\right)}$$Terapkan aturan pangkat $$$\frac{d}{dz} \left(z^{n}\right) = n z^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dz} \left(z\right) = 1$$$:
$$y \cos{\left(y z \right)} {\color{red}\left(\frac{d}{dz} \left(z\right)\right)} = y \cos{\left(y z \right)} {\color{red}\left(1\right)}$$Dengan demikian, $$$\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right) = y \cos{\left(y z \right)}$$$.
Jawaban
$$$\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right) = y \cos{\left(y z \right)}$$$A