Find the derivative function f' for the following functions f.
f(x) = 2/3x+1; a= -1

Verified step by step guidance

1. Step 1: Identify the function f(x) = \frac{2}{3}x + 1. This is a linear function of the form f(x) = ax + b, where a = \frac{2}{3} and b = 1.
2. Step 2: Recall that the derivative of a linear function f(x) = ax + b is simply the coefficient of x, which is a. In this case, the derivative f'(x) will be \frac{2}{3}.
3. Step 3: Since the derivative of a constant is zero, the +1 in the function does not affect the derivative. Therefore, f'(x) = \frac{2}{3}.
4. Step 4: Evaluate the derivative at the given point a = -1. Since the derivative is constant, f'(-1) = \frac{2}{3}.
5. Step 5: Conclude that the derivative function f'(x) is constant and equal to \frac{2}{3} for all x, including at x = -1.