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