Final answer:
The dot product of two differentiable vector functions u(t) and v(t) is differentiable, but the given derivative statement is false because it is missing the product rule component u(t) · v'(t).
Step-by-step explanation:
The question pertains to the differentiability of the dot product of two vector functions and the correctness of a given derivative statement. When you have two differentiable vector functions u(t) and v(t), their dot product is also a differentiable function of t. This is because the dot product operation is a combination of multiplications and additions of the components of these vectors, which are differentiable operations.
However, the statement (u(t) · v(t))' = u'(t) · v(t) is false. The correct derivative of the dot product of two vectors u(t) and v(t) uses the product rule, which in vector calculus is (u(t) · v(t))' = u'(t) · v(t) + u(t) · v'(t). This is due to the fact that both u(t) and v(t) are functions of t and their rates of change u'(t) and v'(t) contribute to the rate of change of their dot product.