(Please don't use calculus in your answers as I'm attempting to learn differentiation through this question. You can use limits; I understand limits. Thanks!)
Consider $y=x^3$:
(Let us consider only the 1st quadrant for now.)
We can clearly see that the slope of the curve is changing; hence, it is a curve. We can clearly see that the curve becomes steeper and steeper as $x$ increases. That means that the slope is increasing. However, how can we mathematically prove it? How do we calculate its slopes, so that we can prove that the slope of the curve is not constant?