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.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. The derivative is often denoted as f'(x) and represents the slope of the tangent line to the function's graph at any given point.

Power Rule

The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) = n*x^(n-1). This rule simplifies the process of differentiation, especially for functions that can be expressed in polynomial form.

Constant Rule

The Constant Rule states that the derivative of a constant is zero. This means that if a function f(x) is a constant value, its rate of change is zero, indicating that the function does not change regardless of the input. This rule is essential when differentiating functions that include constant terms.