Derivative of $$$\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}$$$
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Your Input
Find $$$\frac{d}{dt} \left(\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dt} \left(\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)\right)} = {\color{red}\left(\frac{d}{dt} \left(\sqrt{2} t\right) - \frac{d}{dt} \left(\sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)\right)}$$The derivative of a constant is $$$0$$$:
$$- {\color{red}\left(\frac{d}{dt} \left(\sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right)\right)} + \frac{d}{dt} \left(\sqrt{2} t\right) = - {\color{red}\left(0\right)} + \frac{d}{dt} \left(\sqrt{2} t\right)$$Apply the constant multiple rule $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ with $$$c = \sqrt{2}$$$ and $$$f{\left(t \right)} = t$$$:
$${\color{red}\left(\frac{d}{dt} \left(\sqrt{2} t\right)\right)} = {\color{red}\left(\sqrt{2} \frac{d}{dt} \left(t\right)\right)}$$Apply the power rule $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dt} \left(t\right) = 1$$$:
$$\sqrt{2} {\color{red}\left(\frac{d}{dt} \left(t\right)\right)} = \sqrt{2} {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{dt} \left(\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right) = \sqrt{2}$$$.
Answer
$$$\frac{d}{dt} \left(\sqrt{2} t - \sqrt{- \sqrt{2} \sqrt{\sqrt{5} + 3} - 2}\right) = \sqrt{2}$$$A