What is the Significance of Derivatives in Calculus?

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In summary, a derivative is the slope of a curve at a specific point and is found by assuming that the curve is a straight line between two infinitely close points. It gives the instantaneous rate of change at that point, rather than the average rate between two points. The slope of the secant line between two points on the curve is the average rate of change, while the slope of the tangent at a point is the instantaneous rate of change. The derivative is a limit and is found by finding the difference in values of y2-y1 over x2-x1 as x2 approaches x1.
  • #1
racer
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Hello there

I know that derivative of a function is the tangent to it's curve and that is the limit of the function as H goes to zero but I still can not understand what it means?

For example let's say F(x)=x^2 and F(x) is a function of a fighter jet plane speed, I know that if I take the slope of the secant line that passes through two points of the fighter jet speed, I'd get the average speed between two points, my question is whose value is the tangent? I read that it is the instaneous rate of change at the point itself rather than the average rate between two point, why it is not considered to be an average speed between two points and the second one is zero. what is the difference between X^2 and it's derivative 2x?

Thanks
 
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  • #2
I think that the best way to describe a derivative is to say that it is the slope of the curve at a specific point. I am sure you are familiar with a slope of a straight line, which is just the difference in y between two points on it divided by the difference in x between the two points. However the slope of a curve is not constant, it varies as a function of x. Calculus finds how the close varies by assuming that if two poins are veery close (aka 0 apart), then the path along the curve between them is just a straight line. This means that we can use the same old method to find the slope at that point. This is called the tangent and is the exact slope. If you go any farther than 0 apart, then you just get the average slope, which is called the secant. That is a at least the mathematical way to look at it.

I am sure you also heard of slope being considered as the amount y changes for every change in x. So for example, let's say you know that y=f(x) where y is the displacement at time x. You also know that the velocity is given by change in displacement / change in time, so basically the slope of y=f(x). Now let's say that the displacement is modeled by y=x^2 and you want to know the velocity at time x=2. Since this is not a linear function, you can't just use the pick-any-two-points method to find the slope at x=2. You have to pick two points that are infinitely close to x, such that they are both x. And using calculus you know that the deriative of y=x^2 (which is dy/dx = 2x) will give you the slope at any arbitary point x. So the instantaneous velocity at time x=2 is dy/dx=2(2)=4.

That is basically what a deriative is. If you want more detail on a part of my explanation, just ask. :)

Btw, if F(x) is a function of speed, then the secant should give you the average acceleration. I think you wanted to say that F(x) is the function of displacement vs time.
 
  • #3
so a derivative is instaneous rate of change because the slope touches one point of the curve while the secant line touches couple points but the strange thing, if you subtract y2-y1 over x2-x1 by plugging values you will get average value while if you use algebriac manipulation, you get the slope of the tangent which has a different value than the first one.

I have a question about Derivatives

Take a look at this picture
http://en.wikipedia.org/wiki/Image:Lim-secant.png

When you are taking the difference in values of y2-y1 over x2-x1, you get the difference over the difference which is the slope of the secant, when you take the slope closer and closer until you get close to zero, the slope of the tangent (( the brown line in the picture )) has the same length of the slope of secant and the slope of the tangent has Y2,Y1,X2,X1 but Y2 and X2 doesn't touches the curve of the function, it is outside the curve but does the derivative gives the value of the difference of the Y2-Y1 over X2-X1. I wanted to see if my I am thinking of is true but it needs precise graphing ,I hope you understand my question and thanks for your help man.
 
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  • #4
racer said:
Take a look at this picture
http://en.wikipedia.org/wiki/Image:Lim-secant.png

When you are taking the difference in values of y2-y1 over x2-x1, you get the difference over the difference which is the slope of the secant, when you take the slope closer and closer until you get close to zero, the slope of the tangent (( the brown line in the picture )) has the same length of the slope of secant and the slope of the tangent has Y2,Y1,X2,X1 but Y2 and X2 doesn't touches the curve of the function, it is outside the curve but does the derivative gives the value of the difference of the Y2-Y1 over X2-X1. I wanted to see if my I am thinking of is true but it needs precise graphing ,I hope you understand my question and thanks for your help man.

I think you need to be a bit clearer in your question.

"When you are taking the difference in values of y2-y1 over x2-x1, you get the difference over the difference which is the slope of the secant" --You are finding the difference in Vertical change, and the difference of horizontal change. And yes it is the slope of the secant provided that (x2, y2) is a second point on your curve.

"The slope of the tangent has the same length of the slope of secant" --This makes no sense, slopes do not have "lengths".

"I wanted to see if my I am thinking of is true but it needs precise graphing" --This makes no sense because of grammar.

"but does the derivative gives the value of the difference of the Y2-Y1 over X2-X1" -- Yes, but the derivative is a limit, so think of (x2, y2) being infinitely close to (x1, y1).
 
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  • #5
if0oj8.jpg


By the way, the Y2 of the point is supposed to be Y1 but I mistakenly written Y2.

"When you are taking the difference in values of y2-y1 over x2-x1, you get the difference over the difference which is the slope of the secant" --You are finding the difference in Vertical change, and the difference of horizontal change. And yes it is the slope of the secant provided that (x2, y2) is a second point on your curve.

"The slope of the tangent has the same length of the slope of secant" --This makes no sense, slopes do not have "lengths".

Many Math websites when trying to draw the derivative, they draw a secant line passing through two points and exeeding them and when they draw another one closer and closer, they deliberately draw lines that has the same length while they mean the difference between two points, I thought the derivative as the change of Y's over change of X's and I uploaded a picture that describes what I meant using the cartesian coordinate

"I wanted to see if my I am thinking of is true but it needs precise graphing" --This makes no sense because of grammar.

I wanted to see if what I was thinking of was true or not but I needed precise graphing because I could not draw a precise quadratic function.

"but does the derivative gives the value of the difference of the Y2-Y1 over X2-X1" -- Yes, but the derivative is a limit, so think of (x2, y2) being infinitely close to (x1, y1).

So a derivative means instaneous rate of change and a limit where the change in X gets arbitrarily close to zero
 
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  • #6
You might also want to refer to this thread :smile:
 
  • #7
You might also want to refer to this thread

Thanks, your post is very informative.
 

Related to What is the Significance of Derivatives in Calculus?

1. What is the definition of a derivative?

The derivative of a function at a specific point is the slope of the tangent line to the function at that point.

2. Why is the derivative important in calculus?

The derivative is a fundamental concept in calculus that allows us to analyze the rate of change of a function. It is used to find maximum and minimum values, solve optimization problems, and understand the behavior of functions.

3. How is the derivative calculated?

The derivative is calculated using the limit definition, which takes the difference quotient of the function as the change in x approaches 0. This can also be done using various differentiation rules, such as the power rule, product rule, and chain rule.

4. What is the relationship between the derivative and the slope of a curve?

The derivative of a function at a specific point is equal to the slope of the tangent line to the function at that point. This means that the derivative represents the instantaneous rate of change of a function at a given point.

5. How is the derivative used in real-world applications?

The derivative is used in various real-world applications, such as in physics to analyze motion and acceleration, in economics to calculate marginal cost and revenue, and in engineering to optimize designs and solve problems related to rates of change.

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