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

  1. Mar 26, 2017 #1
    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.
     
  2. jcsd
  3. Mar 26, 2017 #2

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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.
     
  4. Mar 26, 2017 #3
    But at other points the function is the same so wouldn't by power rule the derivative be same??
     
  5. Mar 26, 2017 #4

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    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.
     
  6. Mar 26, 2017 #5
    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.)
     
  7. Mar 26, 2017 #6

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    Yes, you are right. f'(x) = 2x for any value of x.
     
  8. Mar 26, 2017 #7

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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
     
  9. Mar 26, 2017 #8
    So as x changes derivative changes. Thanks.
     
  10. Mar 26, 2017 #9

    Mark44

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    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.
     
  11. Mar 26, 2017 #10
    My bad
     
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