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

In summary: I misunderstood what you were asking. The derivative of 1/x2 is always 1/2x, no matter where x is in the real world. So yes, it will be 2x everywhere.
  • #1
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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  • #2
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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  • #3
But at other points the function is the same so wouldn't by power rule the derivative be same??
 
  • #4
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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  • #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.)
 
  • #6
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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  • #7
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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  • #8
So as x changes derivative changes. Thanks.
 
  • #9
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.)
 
  • #10
My bad
 

1. What is the derivative of a function?

The derivative of a function is a measure of how the function changes at a specific point. It represents the slope of the tangent line to the curve at that point.

2. Is the derivative of a function the same at every point on a curve?

No, the derivative of a function can vary at different points on a curve. It depends on the slope of the tangent line at each specific point.

3. Can the derivative of a function be negative?

Yes, the derivative of a function can be negative if the slope of the tangent line at that point is downward, indicating a decrease in the function's value.

4. How do you find the derivative of a function?

The derivative of a function can be found using the derivative rules, such as the power rule or the product rule. It involves finding the limit of the slope of the secant line as the two points on the curve get closer together.

5. Why is the derivative important in calculus?

The derivative is important in calculus because it helps us understand how a function changes at a specific point, which is crucial in many real-world applications. It is also used to find the maximum and minimum values of a function, and to solve optimization problems.

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