Evaluate

 I had a question from someone who was taught that the

inflexion point of a quadratic equation was -b/2a, and why was

I making a fuss about 'solving for the first derivative'... 

Obviously, they are the same.

Indeed I am not the one making the fuss, but the issue arises in the

history of mathematics, and the emergence of Calculus with Newton

and Leibniz.

Part of the problem with the derivative is the obscurantist language in its

formulization. What is this business about the slope of a point. Since when

do points have slope. A more modern way of putting it would be that - in

the case of a quadratic equation - the x-axis represents linear values, but the

y-axis areas ie squares. So that as x increases, y increases more quickly. Thus,

leading one to question what that relationship might be at any point in time.

So here is where the first derivative is of interest. It is a line that shows this 

relationship. The derivative of X^2 is 2x, always. Below, how someone from

a help forum explained it:

                                                                  

Below, (x + 2)(x + 3) = x^2 + 5x + 6; which I entered in Graph. The first derivative

function is 2x + 5, which solved as 2x + 5 = 0 gives -2.5. And that is -b/2a!!

Either side of this - the point of inflextion - the function evolves at an analogous speed.

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