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

[Debaa]

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.

 

[FactChecker]

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.

 

[Debaa]

But at other points the function is the same so wouldn't by power rule the derivative be same??

 

[FactChecker]

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.

 

[Debaa]

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.)

 

[FactChecker]

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.

 

[FactChecker]

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.

 

[Debaa]

So as x changes derivative changes. Thanks.

 

[Mark44]

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.

And it's derivative is always 2x isn't it? Will it be 2x everywhere? (I am new to calculus so pls bare.)

 

[Debaa]

My bad