Final answer:
The derivative of the vector function r(t) = 4e^ti + 5e^-tj, denoted as r'(t), is 4e^ti - 5e^-tj, providing the velocity vector for any time t.
Step-by-step explanation:
To find r'(t), we need to calculate the derivative of the vector function r(t) = 4eti + 5e-tj with respect to t. The derivative of a vector function is found by differentiating each of the component functions individually.
The derivative of the i component, which is 4et, is simply 4et, because the derivative of e to the power of t with respect to t is et.
The derivative of the j component, which is 5e-t, involves the chain rule. The derivative of e to the power of -t is -e-t, and hence we multiply this by the coefficient 5 to get -5e-t.
Therefore, r'(t) = 4eti - 5e-tj. This gives us the velocity vector at any time t.