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Find $$$\frac{d^{2}}{dt^{2}} \left(5 \cos{\left(t \right)}\right)$$$
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Find $$$\frac{d^{2}}{dt^{2}} \left(5 \cos{\left(t \right)}\right)$$$.
Solution
Find the first derivative $$$\frac{d}{dt} \left(5 \cos{\left(t \right)}\right)$$$
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 $$$c = 5$$$ and $$$f{\left(t \right)} = \cos{\left(t \right)}$$$:
$${\color{red}\left(\frac{d}{dt} \left(5 \cos{\left(t \right)}\right)\right)} = {\color{red}\left(5 \frac{d}{dt} \left(\cos{\left(t \right)}\right)\right)}$$The derivative of the cosine is $$$\frac{d}{dt} \left(\cos{\left(t \right)}\right) = - \sin{\left(t \right)}$$$:
$$5 {\color{red}\left(\frac{d}{dt} \left(\cos{\left(t \right)}\right)\right)} = 5 {\color{red}\left(- \sin{\left(t \right)}\right)}$$Thus, $$$\frac{d}{dt} \left(5 \cos{\left(t \right)}\right) = - 5 \sin{\left(t \right)}$$$.
Next, $$$\frac{d^{2}}{dt^{2}} \left(5 \cos{\left(t \right)}\right) = \frac{d}{dt} \left(- 5 \sin{\left(t \right)}\right)$$$
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 $$$c = -5$$$ and $$$f{\left(t \right)} = \sin{\left(t \right)}$$$:
$${\color{red}\left(\frac{d}{dt} \left(- 5 \sin{\left(t \right)}\right)\right)} = {\color{red}\left(- 5 \frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)}$$The derivative of the sine is $$$\frac{d}{dt} \left(\sin{\left(t \right)}\right) = \cos{\left(t \right)}$$$:
$$- 5 {\color{red}\left(\frac{d}{dt} \left(\sin{\left(t \right)}\right)\right)} = - 5 {\color{red}\left(\cos{\left(t \right)}\right)}$$Thus, $$$\frac{d}{dt} \left(- 5 \sin{\left(t \right)}\right) = - 5 \cos{\left(t \right)}$$$.
Therefore, $$$\frac{d^{2}}{dt^{2}} \left(5 \cos{\left(t \right)}\right) = - 5 \cos{\left(t \right)}$$$.
Answer
$$$\frac{d^{2}}{dt^{2}} \left(5 \cos{\left(t \right)}\right) = - 5 \cos{\left(t \right)}$$$A