Derivatives of Curves. How do they work? How is the second derivative of a quadratic equation a constant? Is the second derivative the slope of the slope?
In: Mathematics
By taking the derivative of a function, we are essentially looking at the rate of change. Let’s take a simple example: f(x) = x^(2) – x + 1
If we take the first derivative, we end up with f'(x) = 2x + 1
If we plot both functions you’ll see that when f'(x) is equal to 0. The change in f(x) is also 0. Because the parabola has a minimum, at this point, it doesn’t go lower or higher. You can see this by looking at the green arrow in my [plot](http://prntscr.com/10n6ihw).
Before the minimum, if we move along the x-axis (left to right), we see that the parabola is decreasing. This is also indicated by f'(x) because the linear equation is negative at this point. Vice versa, if we move beyond the parabola’s minimum, it will start to increase. Again, here its derivative is positive.
Now for simplicity we will make the first derivative it’s own function! so f'(x) = g(x). Taking the derivative of this will indeed result in a constant. Why? Because the rate of change of g(x) is also constant. If we look at a new [plot](http://prntscr.com/10n6n1l) we see that that g(x) = 2x -1 increases by 2 y units, for every whole unit of x. This is also supported by its derivative g'(x) = 2
So when we take the second derivative of a function, we first look at the rate of change, and then we look at the rate of change OF the rate of change.
It’s more difficult to explain by words than to look at the graphs, I hope this helped you visualize it some more.
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