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

Click For Summary
The derivative of a function can vary at different points along a curve, particularly for curved lines, where the slope changes. While a straight line has a constant slope, curved functions like f(x) = x^2 have derivatives that are consistent in form (2x) but vary in value depending on x. It's important to note that not all functions have simple derivative equations, and some can exhibit complex behavior. The discussion also clarifies that the graph of f(x) = x^2 does not have a vertical asymptote, contrasting it with functions like f(x) = 1/x^2, which do. Understanding these concepts is crucial for grasping calculus fundamentals.
Debaa
Messages
22
Reaction score
0
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.
 
Physics news on Phys.org
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.
 
  • Like
Likes Debaa
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.
 
  • Like
Likes 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.)
 
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.
 
  • Like
Likes Debaa
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
 
  • Like
Likes Debaa
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.)
 
  • #10
My bad
 

Similar threads

Undergrad Partial derivatives of the function f(x,y)
  • · Replies 6 ·
Replies
6
Views
2K
High School Question about units for "area under curve"
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Gradient Vectors: Perpendicular to Level Curves
  • · Replies 4 ·
Replies
4
Views
2K
High School Tangents to a curve as functions of different variables
  • · Replies 5 ·
Replies
5
Views
2K
Graduate How to take the spatial derivative of quaternions
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Video (analytic continuation) seems to mix 4-D & 2-D maps
  • · Replies 8 ·
Replies
8
Views
2K
High School Why the derivative of area is related to the graph of the function
  • · Replies 12 ·
Replies
12
Views
3K
High School What does the derivative of a function at a point describe?
  • · Replies 36 ·
2
Replies
36
Views
4K
Undergrad What would a calculus author have to say on ##\int r^2dm##?
  • · Replies 11 ·
Replies
11
Views
3K
Undergrad Taking the derivative of a function
  • · Replies 15 ·
Replies
15
Views
3K