Answer
(a) The sum rule (differentiation rule of vector function):
$\frac{d}{dt}[u(t)+v(t)]=u'(t)+v'(t)$
(b) The scalar multiple rule ( differentiation rule of vector function):
$\frac{d}{dt}[cu(t)]=cu'(t)$
(c) The product rule ( differentiation rule of vector function):
$\frac{d}{dt}[f(t)u(t)]=f'(t)u(t)+f(t)u'(t)$
(d) The dot-product rule ( differentiation rule of vector function):
$\frac{d}{dt}[f(t) \cdot u(t)]=f'(t) \cdot u(t)+f(t) \cdot u'(t)$
(e) The cross-product rule ( differentiation rule of vector function):
$\frac{d}{dt}[f(t) \times u(t)]=f'(t) \times u(t)+f(t) \times u'(t)$
(f) The chain rule ( differentiation rule of vector function):
$\frac{d}{dt}[u(f(t))]=f'(t)u'(f(t))$
Work Step by Step
(a) The sum rule (differentiation rule of vector function):
$\frac{d}{dt}[u(t)+v(t)]=u'(t)+v'(t)$
(b) The scalar multiple rule ( differentiation rule of vector function):
$\frac{d}{dt}[cu(t)]=cu'(t)$
(c) The product rule ( differentiation rule of vector function):
$\frac{d}{dt}[f(t)u(t)]=f'(t)u(t)+f(t)u'(t)$
(d) The dot-product rule ( differentiation rule of vector function):
$\frac{d}{dt}[f(t) \cdot u(t)]=f'(t) \cdot u(t)+f(t) \cdot u'(t)$
(e) The cross-product rule ( differentiation rule of vector function):
$\frac{d}{dt}[f(t) \times u(t)]=f'(t) \times u(t)+f(t) \times u'(t)$
(f) The chain rule ( differentiation rule of vector function):
$\frac{d}{dt}[u(f(t))]=f'(t)u'(f(t))$
