To find the derivative of the vector function r(t) = a + 4b + t^5c, where a, b, and c are constant vectors, we can differentiate each component of the vector separately.
Let's break down the vector function into its components:
r(t) = (a_x + 4b_x + t^5c_x, a_y + 4b_y + t^5c_y, a_z + 4b_z + t^5c_z)
To find the derivative, we differentiate each component with respect to t:
r'(t) = (d/dt (a_x + 4b_x + t^5c_x), d/dt (a_y + 4b_y + t^5c_y), d/dt (a_z + 4b_z + t^5c_z))
Since a, b, and c are constant vectors, their derivatives with respect to t are zero:
r'(t) = (0 + 0 + 5t^4c_x, 0 + 0 + 5t^4c_y, 0 + 0 + 5t^4c_z)
Therefore, the derivative of the vector function r(t) = a + 4b + t^5c is:
r'(t) = (5t^4c_x, 5t^4c_y, 5t^4c_z)