The derivative without time.

In summary, the derivative of the area of a circle with respect to the radius is 2*pi*r, which represents the rate of change of the area as the radius increases. Similarly, the derivative of the area of a square with respect to its side length is the perimeter, or 4*x. These derivatives do not necessarily involve time, but they measure the rate of change of the area with respect to changes in the given variable. In the example of a circle expanding uniformly over time, the derivative of the area with respect to time can be calculated by multiplying the derivative of the radius with respect to time by 2*pi*r.
  • #1
jasonlr82794
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So I know that the derivative of the area of a circle is 2∏r or the circumference, and that the derivative of the area of a square is the perimeter or 2x. I don't get how these are the derivatives because there isn't a time involved. I thought that the derivative measured the rate of change and if so how do these derivatives without time measure this change?
 
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  • #2
measured the rate of change
Rate of change with respect to ________

You can fill in the blank with anything. For example, as you go up in an airplane the temperature of the air around you gets colder. You can have a function T(h) which is the temperature at a height of h, and T'(h) is the rate of change of the temperature with respect to height - it tells you approximately how much the temperature changes if you go one meter higher.

Similiarly, the derivative of the area of the circle with respect to the radius tells you how much the area of the circle increases by if you increase the radius of the circle
 
  • #3
Ok so how would you write down the rate of change of the area of a circle. for example it would be two miles per hour but what would it be for the circle. I hope this makes some sense. and thank you for replying. It seems nobody has replied on my other posts.
 
  • #4
jasonlr82794 said:
Ok so how would you write down the rate of change of the area of a circle. for example it would be two miles per hour but what would it be for the circle. I hope this makes some sense. and thank you for replying. It seems nobody has replied on my other posts.

Assuming that all the units are in meters for example, the derivative of the area with respect to radius would be meters squared per meter. The units of a derivative are always units of the value of the function divided by units of the input of the function
 
  • #5
Isaac Newton considered derivatives to be with respect to time, only. His notation for the derivative time derivative of x looked like this: ## \dot{x}##, while Leibniz used dx/dt to represent the same thing. The "prime" notation that is used in calculus today derives from Newton's dot notation.

For your example of the circle, if both A and r are differentiable functions of t (time), then dA/dt = ##\pi##r2(t), so dA/dt = dA/dr * dr/dt = 2##\pi##r dr/dt.

Here the expression 2##\pi##r is dA/dr, the rate of change of area with respect to change in radius.
 
  • #6
Ok, to office shredder, the radiuses derivative would be meters squared per meters because the radius is the acceleration and acceleration is squared? This is how I came up with this. accelerationxtime=rate, accerlerationxtime=d/t, accerlerationxt^2=distance?
 
  • #7
jasonlr82794 said:
Ok, to office shredder, the radiuses derivative would be meters squared per meters because the radius is the acceleration and acceleration is squared?
No. Assuming you mean the rate of change of the radius with respect to time, dr/dt, the units would be meters/seconds or meters/minutes, or whatever the units of time are.
jasonlr82794 said:
This is how I came up with this. accelerationxtime=rate, accerlerationxtime=d/t, accerlerationxt^2=distance?
This is pretty mixed up. Assuming length units of meters, and time units of seconds,
acceleration = meters2/seconds2
velocity = meters/seconds
 
  • #8
Im not getting the example you had because I don't want it to have a function of time. But would this be true...
A=∏r^2
A primed= 2∏r
So how would you put this into a function of time?
 
  • #9
The perimeter of a square with side x is 4*x, not 2*x.
 
  • #10
jasonlr82794 said:
Im not getting the example you had because I don't want it to have a function of time. But would this be true...
A=∏r^2
A primed= 2∏r
So how would you put this into a function of time?

He was explaining how there is no time involved with taking that derivative, but if you have the radius as a function of time then you can alternatively find the derivative of area with respect to time
 
  • #11
jasonlr82794 said:
Im not getting the example you had because I don't want it to have a function of time. But would this be true...
A=∏r^2
A primed= 2∏r
So how would you put this into a function of time?

Think of a circle that expands, or contracts, uniformly over time. Then, the rate of change of the area over time will be 2*pi*r*(dr/dt), so that if the the radial increase pr.time unit Equals one distance unit, you will have dA/dt=2*pi*r
 

1. What is the derivative without time?

The derivative without time is a mathematical concept that refers to the instantaneous rate of change of a function at a specific point, without considering the element of time. It is a fundamental concept in calculus and is used to analyze the behavior of a function at a particular point.

2. How is the derivative without time calculated?

The derivative without time is calculated using the limit definition of a derivative, which involves taking the limit of the change in the function over the change in the independent variable as the change approaches zero. It can also be calculated using differentiation rules or by using software such as WolframAlpha.

3. What is the difference between the derivative with time and without time?

The main difference between the derivative with time and without time is that the derivative with time considers the element of time and measures the rate of change over a specific interval, while the derivative without time measures the instantaneous rate of change at a single point in time.

4. What are some real-life applications of the derivative without time?

The derivative without time has numerous real-life applications, including predicting and analyzing stock market trends, optimizing production processes in industries, and determining the maximum and minimum values of a function in physics and engineering problems.

5. Are there any limitations or restrictions when using the derivative without time?

Yes, there are some limitations or restrictions when using the derivative without time. It is not applicable to all functions, and the function must be continuous and differentiable at the point of interest. Additionally, the derivative without time may not accurately represent the behavior of a function over a larger interval, as it only measures the instantaneous rate of change at a single point.

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