What is the d/(dt)(tcos(t)) ?

The d/(dt)(tcos(t)) is cos(t)-tsin(t)

What is the first d/(dt)(tcos(t)) ?

The first d/(dt)(tcos(t)) is cos(t)-tsin(t)

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\frac{d}{d t} (t cos (t))

cos (t) - t sin (t)

Solution steps

\frac{d}{d t} (t cos (t))

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = t, g = cos (t)

= \frac{d t}{d t} cos (t) + \frac{d}{d t} (cos (t)) t

\frac{d t}{d t} = 1

\frac{d}{d t} (cos (t)) = - sin (t)

= 1 \cdot cos (t) + (- sin (t)) t

Simplify

= cos (t) - t sin (t)