Is the derivative of a function everywhere the same on a given curve?

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Discussion Overview

The discussion revolves around the nature of derivatives of functions along curves, specifically whether the derivative is constant across a curve or varies at different points. It touches on concepts from calculus, including the power rule and the behavior of derivatives for different types of functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the derivative of a function is the same everywhere on a curve or only in infinitesimally small segments.
  • Another participant asserts that the derivative can differ at various points on a curve, noting that a straight line has a constant slope while a curved line does not.
  • A participant suggests that if a function has the same equation at different points, the derivative should also be the same, referencing the power rule.
  • Clarification is provided that while a function may have the same equation, the value of the derivative can vary at different points, especially for functions without a simple equation form.
  • One participant mentions the function f(x) = x^2, questioning its properties and whether its derivative, 2x, is consistent across all x values.
  • Another participant confirms that the derivative of f(x) = x^2 is indeed 2x for every value of x, providing a graphical reference to support this claim.
  • There is a correction regarding the term "vertically asymptotic," with a participant clarifying that the graph of y = x^2 does not have a vertical asymptote, contrasting it with y = 1/x^2, which does.

Areas of Agreement / Disagreement

Participants express differing views on the behavior of derivatives along curves, with some asserting that derivatives can vary while others focus on specific functions like f(x) = x^2. The discussion remains unresolved regarding the implications of these differing perspectives.

Contextual Notes

There are limitations in the discussion regarding the definitions of terms like "vertically asymptotic" and the assumptions about the behavior of derivatives for different types of functions.

Debaa
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Is the derivative of a function everywhere the same on a given curve? Or is it just for a infinitesimally small part of the curve? Thank you for the answer.
 
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It can be different for each infinitesimally small part of the curve. (It might be the same but not necessarily so.) The derivative at a point on the curve is the slope of the curve at that point. A straight line has the same slope everywhere. A curved line has different slopes (i.e. derivative) at different points.
 
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But at other points the function is the same so wouldn't by power rule the derivative be same??
 
I should be more clear. The value of the derivative can be different at different points. It may have the same equation, but not the same value. A lot of functions and their derivatives do not have a nice equation form, so you should not count on that in general.
 
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Well f(x) = x^2 is vertically asymptotic. And it's derivative is always 2x isn't it? Will it be 2x everywhere? (I am new to calculus so pls bare.)
 
Debaa said:
Well f(x) = x^2 is vertically asymptotic. And it's derivative is always 2x isn't it? Will it be 2x everywhere? (I am new to calculus so pls bare.)
Yes, you are right. f'(x) = 2x for any value of x.
 
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Yes, you are right. The derivative is 2x for every value of x.
You can see in the following graph that for every value of x, the slope of f(x)=x2 is d(x)=2x.
temp.png
 
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So as x changes derivative changes. Thanks.
 
Debaa said:
Well f(x) = x^2 is vertically asymptotic.
Not sure what you mean by this. The graph of y = x2 does not have a vertical asymptote, although as x gets larger (or more negative) the y value gets larger.

The graph of y = 1/x2 does have a vertical asymptote at x = 0.
Debaa said:
And it's derivative is always 2x isn't it? Will it be 2x everywhere? (I am new to calculus so pls bare.)
 
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My bad
 

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