Requesting steps please :)

Would someone please be able to explain how:

Did you cook this one up in your head? It is FALSE!.

This is how the textbook displays it:

Define the associated gradient function, g(x), for each f(x) given in the table:

It is approximating a derivative:

((x + 0.01)² - x²)/0.01 → (x² + 0.02x + 0.0001 - x²)/0.01 → 2x + 0.01

Now if 2x is much bigger than 0.01, then the above is just 2x.  If you replace the 0.01 by, say 0.0001, and repeat the process, the result is even closer to just 2x.  When you are finding the derivative you replace the 0.01 by, say, h, so you get 2x + h.  You then take the limit as h goes to zero, to get the derivative of x² as exactly 2x.