What is $\frac{d}{dt}x\cos(y)$?

Question

$\dfrac{d}{dt}x = \dot{x}$ and $\dfrac{d}{dt}y = \dot{y}$

but how do I differentiate something like this with respect to time?

$\dfrac{d}{dt}x\cos \left(y\right) = ?$

Answer 1

$\begin{align} \frac{d}{dt}x\cos(y) & = \left(\frac{d}{dt}x\right)\cos y + x\left( \frac{d}{dt}\cos y\right) \\ &= \dot x\cos y - x\dot y\sin y \end{align}$

Answer 2

Better to label $ x \cos y, $ say as $z:$

$$ \frac{dz}{dt} = \frac{dx}{dt} \cos y + x\frac{d\, \cos y }{dt} = \dot x\cos y - x\dot y\sin y. $$