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