A value in the range for any base polynomial function with a y-intercept of zero can be expressed as: $f\left(x\right) = px$ where $p$ is the average rate of change between $0$ and $x$. The average rate of change can be in turn expressed as $p = \frac{ f'\left(0\right) + f'\left(x\right)}{2}$ where $f'\left(0\right)$ is the instantaneous rate of change at $0$ (or in other words the derivative evaluated at zero) and $f'\left(x\right)$ is the instantaneous rate of change evaluated at $x$ (or in other words the derivative evaluated at the specific range value). In a base polynomial equation (or in other words a polynomial with one term of degree $d$ and leading coefficient $1$) with degree greater than 1 the derivative evaluated at zero can be further simplified to $0$. This leads to a final simplified equation: $f(x) = \frac{f'(x)x}{2}$
This final equation however fails to provide the range value for a polynomial above degree $2$. According to the power rule, if: $f(x) = x^3$ then: $f'(x) = 3x^2$ According to the above derived equation:
$f(3) = \frac{f'(3)(3)}{2} = \frac{(3(3)^2)3}{2} = \frac{81}{2}$
which is clearly the wrong answer. What is the flaw in the equation relating the average change of rate and the derivative?