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Derivative of $$$5 \cos{\left(2 t \right)}$$$
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Your Input
Find $$$\frac{d}{dt} \left(5 \cos{\left(2 t \right)}\right)$$$.
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 $$$c = 5$$$ and $$$f{\left(t \right)} = \cos{\left(2 t \right)}$$$:
$${\color{red}\left(\frac{d}{dt} \left(5 \cos{\left(2 t \right)}\right)\right)} = {\color{red}\left(5 \frac{d}{dt} \left(\cos{\left(2 t \right)}\right)\right)}$$The function $$$\cos{\left(2 t \right)}$$$ is the composition $$$f{\left(g{\left(t \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ and $$$g{\left(t \right)} = 2 t$$$.
Apply the chain rule $$$\frac{d}{dt} \left(f{\left(g{\left(t \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dt} \left(g{\left(t \right)}\right)$$$:
$$5 {\color{red}\left(\frac{d}{dt} \left(\cos{\left(2 t \right)}\right)\right)} = 5 {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dt} \left(2 t\right)\right)}$$The derivative of the cosine is $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$$5 {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dt} \left(2 t\right) = 5 {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dt} \left(2 t\right)$$Return to the old variable:
$$- 5 \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dt} \left(2 t\right) = - 5 \sin{\left({\color{red}\left(2 t\right)} \right)} \frac{d}{dt} \left(2 t\right)$$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 $$$c = 2$$$ and $$$f{\left(t \right)} = t$$$:
$$- 5 \sin{\left(2 t \right)} {\color{red}\left(\frac{d}{dt} \left(2 t\right)\right)} = - 5 \sin{\left(2 t \right)} {\color{red}\left(2 \frac{d}{dt} \left(t\right)\right)}$$Apply the power rule $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dt} \left(t\right) = 1$$$:
$$- 10 \sin{\left(2 t \right)} {\color{red}\left(\frac{d}{dt} \left(t\right)\right)} = - 10 \sin{\left(2 t \right)} {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{dt} \left(5 \cos{\left(2 t \right)}\right) = - 10 \sin{\left(2 t \right)}$$$.
Answer
$$$\frac{d}{dt} \left(5 \cos{\left(2 t \right)}\right) = - 10 \sin{\left(2 t \right)}$$$A