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(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 $y=x^3$:

enter image description here

(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?

tryingtobeastoic
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  • What do you know about derivatives? – Alan Abraham Oct 05 '21 at 06:39
  • $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$

    $f'(x)$ is the derivative of $f(x)$. However, my problem with derivatives is that how do we know that they are the best constant approximation, or is it just that they are defined that way? @AlanAbraham

     – tryingtobeastoic Oct 05 '21 at 06:40
  • Yes, that is the definition of the derivative. Have you tried applying it to your problem? – Alan Abraham Oct 05 '21 at 06:41
  • May have another picture: Newton forward and backward difference discussed here. – Ng Chung Tak Oct 05 '21 at 06:44
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    Derivative considerations unnecessary. Consider the line segment from the origin to any point on the curve, in the first quadrant. Notice that the tangent (i.e. slope of the curve) at that point is always steeper than the corresponding line segment. As $x$ increases, the steepness of the corresponding line segment is strictly increasing. – user2661923 Oct 05 '21 at 07:32
  • @user2661923 yeah that is exactly what I'm asking. – tryingtobeastoic Oct 05 '21 at 07:38
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    Re previous comment, one of the assertions is verified by consideration of $\frac{f(x)}{x}.$ The other assertion requires consideration of the nature of convex functions (i.e. upward curving functions). Analytical consideration of convex functions may require consideration of the derivative. – user2661923 Oct 05 '21 at 07:39
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    Why this downvoting ? It is a well written question with a good dialog with the asker ! – Jean Marie Oct 05 '21 at 09:14

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