How Does Differentiation Determine the Exact Slope of a Curve?

[lazyaditya]

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"?

 

[Number Nine]

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

 

[NascentOxygen]

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.

 

[lazyaditya]

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

 

[NascentOxygen]

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

 

[lazyaditya]

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 ?

 

[Number Nine]

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.

 

[NascentOxygen]

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.

 

[lazyaditya]

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

 

[Number Nine]

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.

 

[HallsofIvy]

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!

 

[NascentOxygen]

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.

 

[lazyaditya]

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 :)