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Derivative of $$$e^{t} \cos{\left(t \right)}$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right)$$$.
Solution
Apply the product rule $$$\frac{d}{dt} \left(f{\left(t \right)} g{\left(t \right)}\right) = \frac{d}{dt} \left(f{\left(t \right)}\right) g{\left(t \right)} + f{\left(t \right)} \frac{d}{dt} \left(g{\left(t \right)}\right)$$$ with $$$f{\left(t \right)} = \cos{\left(t \right)}$$$ and $$$g{\left(t \right)} = e^{t}$$$:
$${\color{red}\left(\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right)\right)} = {\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)}\right) e^{t} + \cos{\left(t \right)} \frac{d}{dt} \left(e^{t}\right)\right)}$$The derivative of the cosine is $$$\frac{d}{dt} \left(\cos{\left(t \right)}\right) = - \sin{\left(t \right)}$$$:
$$e^{t} {\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)}\right)\right)} + \cos{\left(t \right)} \frac{d}{dt} \left(e^{t}\right) = e^{t} {\color{red}\left(- \sin{\left(t \right)}\right)} + \cos{\left(t \right)} \frac{d}{dt} \left(e^{t}\right)$$The derivative of the exponential is $$$\frac{d}{dt} \left(e^{t}\right) = e^{t}$$$:
$$- e^{t} \sin{\left(t \right)} + \cos{\left(t \right)} {\color{red}\left(\frac{d}{dt} \left(e^{t}\right)\right)} = - e^{t} \sin{\left(t \right)} + \cos{\left(t \right)} {\color{red}\left(e^{t}\right)}$$Simplify:
$$- e^{t} \sin{\left(t \right)} + e^{t} \cos{\left(t \right)} = \sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}$$Thus, $$$\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right) = \sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}$$$.
Answer
$$$\frac{d}{dt} \left(e^{t} \cos{\left(t \right)}\right) = \sqrt{2} e^{t} \cos{\left(t + \frac{\pi}{4} \right)}$$$A