Why is the change in y different than expected when using derivatives?

[Line]

I'm trying to understand something. Derrivatives are basically the same as differentials. The both apply to the chnange in XY values.

Now I have an eqaition y=x*x x*x=x sqaured.

So that being here are my XY values

X y
_______
1 1
2 4
3 9
4 16
5 25


So that being dv=2x . But if you plug that in x the change is different.

WIth that eqauition if x=1 the change in y should be 2. So if x changes by 1. But is chnage x by 1 y becomes 4.

If x=2 the change in y should be 4. But if we change x by 1 y=9 not 8.

What's going on?

 

[d_leet]

Because a derivative is an instantaneous rate of change, and changing x by 1 is a large enough change for the tangent line at the point to not be very close to the original function anymore.

 

[Line]

So what is the derivative for? You can't just plug it in at any point and get the change for y?

 

[d_leet]

So what is the derivative for? You can't just plug it in at any point and get the change for y?

You can, BUT it is an instantaneous rate of change that is only exact at that point. If you try to use that derivative to find other points on the curve it will fail as the deviation from your original x increases, in other words let's say you have

y = f(x)

then
y' = f'(x)

and the differential is

dy = f'(x)dx

this allows you to approximate the original function by knowing its derivative and an initial point. But the further you go from the initial point, as dx increases, the approximation will become less valid because a derivative is an instantaneous rate of change and will be exact only at that point.

 

[Hurkyl]

So that being here are my XY values

X y
_______
1 1
2 4
3 9
4 16
5 25


So that being dv=2x . But if you plug that in x the change is different.

WIth that eqauition if x=1 the change in y should be 2. So if x changes by 1. But is chnage x by 1 y becomes 4.

If x=2 the change in y should be 4. But if we change x by 1 y=9 not 8.

What's going on?

Let's experiment! What if, instead of increasing all of your x's by 1, you increased them by 0.1? So your table would look like:


X y
_______
1 ?
1.1 ?
1.2 ?
1.3 ?
1.4 ?


What if you did it by 0.01?


How do the results compare to the definition of a derivative? (You know, that stuff with limits!)

p.s. if y = x², then dy = 2x dx. (Of course, y' = 2x)