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Turunan dari $$$e^{- x} \sin{\left(x \right)}$$$ pada $$$x = c$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Tentukan $$$\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)$$$ dan hitung nilainya pada $$$x = c$$$.
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 $$$f{\left(x \right)} = e^{- x}$$$ dan $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(e^{- x}\right) \sin{\left(x \right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$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 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$\sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(e^{- x}\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(- x\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Turunan dari fungsi eksponensial adalah $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$$\sin{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)} {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Kembalikan ke variabel semula:
$$e^{{\color{red}\left(u\right)}} \sin{\left(x \right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = e^{{\color{red}\left(- x\right)}} \sin{\left(x \right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$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 $$$f{\left(x \right)} = x$$$:
$$e^{- x} \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(- x\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = e^{- x} \sin{\left(x \right)} {\color{red}\left(- \frac{d}{dx} \left(x\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Turunan fungsi sinus adalah $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$- e^{- x} \sin{\left(x \right)} \frac{d}{dx} \left(x\right) + e^{- x} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = - e^{- x} \sin{\left(x \right)} \frac{d}{dx} \left(x\right) + e^{- x} {\color{red}\left(\cos{\left(x \right)}\right)}$$Terapkan aturan pangkat $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ dengan $$$n = 1$$$, dengan kata lain, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- e^{- x} \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + e^{- x} \cos{\left(x \right)} = - e^{- x} \sin{\left(x \right)} {\color{red}\left(1\right)} + e^{- x} \cos{\left(x \right)}$$Sederhanakan:
$$- e^{- x} \sin{\left(x \right)} + e^{- x} \cos{\left(x \right)} = \sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}$$Dengan demikian, $$$\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right) = \sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}$$$.
Terakhir, hitung nilai turunan pada $$$x = c$$$.
$$$\left(\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)\right)|_{\left(x = c\right)} = \sqrt{2} e^{- c} \cos{\left(c + \frac{\pi}{4} \right)}$$$
Jawaban
$$$\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right) = \sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}$$$A
$$$\left(\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)\right)|_{\left(x = c\right)} = \sqrt{2} e^{- c} \cos{\left(c + \frac{\pi}{4} \right)}\approx 1.414213562373095 e^{- c} \cos{\left(c + \frac{\pi}{4} \right)}$$$A