Answer
a) See the explanation.
b) See the explanation.
c) See the explanation.
d) See the explanation.
e) See the explanation.
f) See the explanation.
Work Step by Step
a)
$u(t)+v(t)=\frac{d}{dt}[u(t)+v(t)]=u'(t)+v'(t)$
It is known as the sum rule of differentiation.
b)
$cu(t)=\frac{d}{dt}[cu(t)]=cu'(t)$
It is known as the scalar multiple rule of differentiation.
(c) $f(t) v(t)=\frac{d}{dt}[f(t)u(t)]=f'(t)u(t)+f(t)u'(t)$
It is known as the product rule of differentiation.
d)
$u(t) \cdot v(t)=\frac{d}{dt}[u(t) \cdot v(t)]=u'(t) \cdot v(t)+u(t) \cdot v'(t)$
It is known as the dot product rule of differentiation.
e)
$u(t) \times v(t)=\frac{d}{dt}[u(t) \times v(t)]=u'(t) \times v(t)+u(t) \times v'(t)$
It is known as the cross product rule of differentiation.
f)
$\frac{d}{dt}[u(f(t))]=f'(t)u'(f(t))$
It is known as the chain rule of differentiation.
