General definition of a derivative

In summary, the general definition of a derivative is f'(x) = limit as delta x approaches 0 of delta y over delta x. However, it cannot work when delta y approaches 0 because the limit is taken on the x-axis. Additionally, f(x) = y or y(x) = y is an abuse of notation and is more commonly used in physics than in mathematics.
  • #1
cscott
782
1
I was told that the general definition of a derivative is

[tex]f'(x) = \lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}[/tex]
(supposed to be delta y over delta x, but I can't make the latex work :mad:)

but why can't it work when [itex]\Delta y \rightarrow 0[/itex]?
 
Last edited by a moderator:
Mathematics news on Phys.org
  • #2
Because the function is [itex] y=y(x) [/itex],so it's natural to consider the limit on the "x" (variable's) axis.


Daniel.
 
  • #3
Oh, alright.

Another thing, what is f(x) = y or y(x) = y in normal notation? I thought f(x) replaced y, but the fuction y = y doesn't make sense, does it? I mixed up :frown:
 
Last edited:
  • #4
Abuse of notation,i don't know how much mathematicians do it,but physicists adore it.

Daniel.
 

1. What is the general definition of a derivative?

The general definition of a derivative is the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point and represents how much the function is changing at that point.

2. How is the general definition of a derivative expressed mathematically?

The general definition of a derivative is expressed as the limit of the difference quotient as the change in the input approaches zero. In other words, it is the limit of the slope of a secant line as the two points on the line get closer and closer together.

3. What is the difference between the general definition of a derivative and the derivative of a specific function?

The general definition of a derivative is a universal concept that applies to all functions, while the derivative of a specific function is the result of applying the general definition to a particular function. The general definition provides a framework for finding the derivative of any function, while the derivative of a specific function is the specific value of the derivative at a given point.

4. How is the general definition of a derivative used in real-world applications?

The general definition of a derivative is used in real-world applications to calculate rates of change and slopes of curves. It is used in physics to calculate velocity and acceleration, in economics to calculate marginal cost and revenue, and in engineering to calculate rates of change in various systems.

5. Are there any limitations to the general definition of a derivative?

While the general definition of a derivative is a powerful concept, there are some limitations to its application. It can only be applied to functions that are continuous and differentiable at a given point. Also, when using the general definition to find the derivative of a function, the limit may not exist if the function has sharp corners or cusps at that point.

Similar threads

  • General Math
Replies
5
Views
959
  • General Math
Replies
2
Views
691
Replies
12
Views
905
Replies
2
Views
2K
Replies
6
Views
3K
  • Linear and Abstract Algebra
Replies
5
Views
911
  • Quantum Physics
Replies
4
Views
1K
  • Advanced Physics Homework Help
Replies
8
Views
714
  • General Math
Replies
3
Views
2K
Back
Top