How Does Differentiation Determine the Exact Slope of a Curve?

  • Context: Undergrad 
  • Thread starter Thread starter lazyaditya
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around the concept of differentiation and its role in determining the slope of a curve. Participants explore the definitions and implications of differentiation, contrasting it with the traditional slope calculation using two points.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that differentiation represents the slope of a curve at a specific point, as indicated by the derivative "dy/dx".
  • Others argue that the traditional slope formula (y2-y1)/(x2-x1) applies to the slope of a line connecting two points, which differs from the instantaneous slope provided by differentiation.
  • A participant questions the significance of using differentiation versus the two-point method for calculating slope.
  • Some participants clarify that differentiation provides an exact slope at a point, while the two-point method yields an approximation.
  • There is a suggestion that understanding the limit process is crucial for grasping the concept of differentiation.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between differentiation and traditional slope calculations. While some agree on the definitions and implications of differentiation, others remain confused about its significance compared to the two-point method.

Contextual Notes

There are unresolved questions regarding the understanding of limits and the conditions under which each method of slope calculation is preferable.

lazyaditya
Messages
176
Reaction score
7
We say that a slope can be calculated as the ratio of the perpendicular to the base , Then what does differentiation of the equation shows "dy/dx"?
 
Physics news on Phys.org


I'm sorry, what? Can you please clarify your question?
 


We say that a slope can be calculated as the ratio of the perpendicular to the base of a line tangential[/color], Then what does differentiation of the equation shows "dy/dx"?
Differential calculus involves the evaluation of that Δy / Δx but taken to its extreme.
 


I just want to know what actually differentiation is apart from its equation ? what does it mean?
 


Differentiation is a way of calculating the slope at any point along a curve if you know that curve's equation.
 


Ya that's what i m saying suppose y =3x^2 then on differentiation i would get (dy/dx=6x) then at point x=3 the slope would be 18. Then isn't slope also calculated as (y2-y1)/(x2-x1) then at point x = 3 isn't it should be 9 ?
 


lazyaditya said:
Ya that's what i m saying suppose y =3x^2 then on differentiation i would get (dy/dx=6x) then at point x=3 the slope would be 18. Then isn't slope also calculated as (y2-y1)/(x2-x1) then at point x = 3 isn't it should be 9 ?

No, because that equation gives you the slope of the line connecting two points; we don't have two points in this case, we only have one. Remember that when we construct the derivative we actually use the above slope formula to calculate the limit of the slope as the distance of two points approaches zero.
 


lazyaditya said:
Then isn't slope also calculated as (y2-y1)/(x2-x1)
Yes. So if you take two points (x1,y1) and (x2,y2) on that curve, where x1 and x2 are very close to 3, then that quotient will be very close to 18.
 


Then what is the significance of one over the other ? I mean in which situation which one should i prefer ?
 
  • #10


lazyaditya said:
Then what is the significance of one over the other ? I mean in which situation which one should i prefer ?

This has already been explained to you. The slope formula gives you the slope of the line connecting two points; the derivative gives the slope of the tangent line at each point of the function.
 
  • #11


What appears to be puzzling you is the limit process. I recommend you go back and review "limits". It's probably in the chapter of your Calculus book just before the derivative is introduced!
 
  • #12


lazyaditya said:
Then what is the significance of one over the other ? I mean in which situation which one should i prefer ?
Differentiation gives the exact value of the slope at any point you choose. Using the "pair of points" method gives an approximation, you are in effect approximating the curve with a straight line.
 
  • #13


NascentOxygen said:
Differentiation gives the exact value of the slope at any point you choose. Using the "pair of points" method gives an approximation, you are in effect approximating the curve with a straight line.

Thank you i was really confused at this :)
 

Similar threads

High School Why does the Limit Process give an Exact Slope?
  • · Replies 53 ·
2
Replies
53
Views
7K
Undergrad Integral confusion for a simple Differential Equation
  • · Replies 2 ·
Replies
2
Views
2K
High School Problem with the concept of differentiation
  • · Replies 46 ·
2
Replies
46
Views
6K
Undergrad Direction Fields and Isoclines
  • · Replies 1 ·
Replies
1
Views
1K
MHB Calculating the Slope of the IS-curve - Hint Included!
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Change of variables clarification
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Is the Equation for Integrating an Exact Differential Correct?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Question about Constrained Differentials
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad Landscape Fabric Roll
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Area Differential in Cartesian and Polar Coordinates
  • · Replies 13 ·
Replies
13
Views
3K