How can I calculate the derivative of the derivative using the tangent line?

Question

How can I calculate the derivative of the derivative of a function $ f(x) $ using the tangent line of a point from that function $ f(x) $ ?

Answer 1

If you ONLY know the line tangent to the graph of a function at some point you CANNOT determine the derivative of the derivative (that is, the second derivative). There are many functions that are tangent to a line at a given point which have different second derivatives. To see this draw a simple line, fix a point and draw many curves tangent to it, the derivative of the derivative is the rate at which the growth (or decrease) rate changes.

For example, the family of functions $x^2,x^4,x^6,...$ they all have the real line $y=0$ as tangent line at the point $x=0$, but their second derivatives are $2x, 4x^3, 6x^5...$ respectively.

However, if you knew the tangent line at EVERY point or at least around some point of interest $p$, then it'd be possible to calculate the second derivative. If $m(x)$ denotes the slope of the tangent line at a point $x$, then the second derivative at a point $p$ is simply (if it exists) $$ \lim_{h \to 0} \frac{m(p+h)-m(p)}{h}$$

EDIT: a better example is given by a comment to this answer below.