Derivative of $$$- \frac{\sqrt{5} \sin{\left(t \right)}}{5}$$$
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Your Input
Find $$$\frac{d}{dt} \left(- \frac{\sqrt{5} \sin{\left(t \right)}}{5}\right)$$$.
Solution
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 $$$c = - \frac{\sqrt{5}}{5}$$$ and $$$f{\left(t \right)} = \sin{\left(t \right)}$$$:
$${\color{red}\left(\frac{d}{dt} \left(- \frac{\sqrt{5} \sin{\left(t \right)}}{5}\right)\right)} = {\color{red}\left(- \frac{\sqrt{5}}{5} \frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)}$$The derivative of the sine is $$$\frac{d}{dt} \left(\sin{\left(t \right)}\right) = \cos{\left(t \right)}$$$:
$$- \frac{\sqrt{5} {\color{red}\left(\frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)}}{5} = - \frac{\sqrt{5} {\color{red}\left(\cos{\left(t \right)}\right)}}{5}$$Thus, $$$\frac{d}{dt} \left(- \frac{\sqrt{5} \sin{\left(t \right)}}{5}\right) = - \frac{\sqrt{5} \cos{\left(t \right)}}{5}$$$.
Answer
$$$\frac{d}{dt} \left(- \frac{\sqrt{5} \sin{\left(t \right)}}{5}\right) = - \frac{\sqrt{5} \cos{\left(t \right)}}{5}$$$A