Turunan dari $$$e^{t} \cos{\left(t \right)}$$$
Kalkulator terkait: Kalkulator Diferensiasi Logaritmik, Kalkulator Diferensiasi Implisit dengan Langkah-langkah
Masukan Anda
Temukan $$$\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right)$$$.
Solusi
Terapkan aturan hasil kali $$$\frac{d}{dt} \left(f{\left(t \right)} g{\left(t \right)}\right) = \frac{d}{dt} \left(f{\left(t \right)}\right) g{\left(t \right)} + f{\left(t \right)} \frac{d}{dt} \left(g{\left(t \right)}\right)$$$ pada $$$f{\left(t \right)} = \cos{\left(t \right)}$$$ dan $$$g{\left(t \right)} = e^{t}$$$:
$${\color{red}\left(\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right)\right)} = {\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)}\right) e^{t} + \cos{\left(t \right)} \frac{d}{dt} \left(e^{t}\right)\right)}$$Turunan fungsi kosinus adalah $$$\frac{d}{dt} \left(\cos{\left(t \right)}\right) = - \sin{\left(t \right)}$$$:
$$e^{t} {\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)}\right)\right)} + \cos{\left(t \right)} \frac{d}{dt} \left(e^{t}\right) = e^{t} {\color{red}\left(- \sin{\left(t \right)}\right)} + \cos{\left(t \right)} \frac{d}{dt} \left(e^{t}\right)$$Turunan dari fungsi eksponensial adalah $$$\frac{d}{dt} \left(e^{t}\right) = e^{t}$$$:
$$- e^{t} \sin{\left(t \right)} + \cos{\left(t \right)} {\color{red}\left(\frac{d}{dt} \left(e^{t}\right)\right)} = - e^{t} \sin{\left(t \right)} + \cos{\left(t \right)} {\color{red}\left(e^{t}\right)}$$Sederhanakan:
$$- e^{t} \sin{\left(t \right)} + e^{t} \cos{\left(t \right)} = \sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}$$Dengan demikian, $$$\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right) = \sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}$$$.
Jawaban
$$$\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right) = \sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}$$$A