Suppose you have a function f(x) = 4x^3 + 3x^2 + 5x + 2.  What is the slope function and how do can you check that your slope function is correct?

The derivative of f(x), f'(x) = 12x^2 + 6x + 5.

We can check by taking the derivative using the slope function at a point, say x = 1.  The derivative of that is 23.

Now use the mean value theorem, [f(b) - f(a)]/(b - a).  Use values for a and b very close to 0, one on either side to approximate the derivative at 0.

a =1.1
b =0.9

f(0.9) = 11.846
f(1.1) = 16.454

Therefore, we get (11.846 - 16.454)/(0.9 - 1.1)  = 23.04  so the slope function holds true.