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

1. Mar 26, 2017

### 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.

2. Mar 26, 2017

### 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.

3. Mar 26, 2017

### Debaa

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

4. Mar 26, 2017

### 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.

5. Mar 26, 2017

### 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.)

6. Mar 26, 2017

### FactChecker

Yes, you are right. f'(x) = 2x for any value of x.

7. Mar 26, 2017

### 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.

8. Mar 26, 2017

### Debaa

So as x changes derivative changes. Thanks.

9. Mar 26, 2017

### Staff: Mentor

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.

10. Mar 26, 2017