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
A derivative of a curve, y, evaluated at a point along the x-axis, x, is the slope of the line tangent to y at x.
When you calculate the derivative as a function it is just a quick way to find the slope of a tangent line at any point.
The second derivative is, indeed, the slope of lines tangent to points along the first derivative. Since the derivative of x^2 is just the line 2x passing through the origin, a tangent line at any point will just be the same line with a slope of 2.
Since this second derivative is just a constant, a horizontal line on the x-y plane, its tangent is just 0. It has no slope at any point.
A fourth derivative of a quadratic and beyond would all just be taking the slope of the horizontal line following the x axis and are mutually 0.
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