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1. Hi, I'm in Year 11 doing my GCSEs. Today in class we looked at differentiation, instantaneous rates of change and average rates of change. The actual process was nothing new as I have done it before in my spare time, however I have a few questions about why and how we do it:

Is there a difference between instantaneous and average rates of change?

What does "derivative" actually mean?

Why do we only differentiate once (in GCSE)? Why not twice? Actually, why do we do it at all? I mean, I know the purpose is to find the rate at which y changes in respect to x (I think), but why can't we just substitute into the original f(x) instead of differentiating?

What does f'(x) mean? What does the apostrophe represent?

Sorry for so many questions. I've been told that at GCSE we only need to know how to do the process rather than knowing/understanding what we're actually doing, but I'd prefer to understand as much as possible about it. Thanks in advance.
2. (Original post by irrelevant kid)
Hi, I'm in Year 11 doing my GCSEs. Today in class we looked at differentiation, instantaneous rates of change and average rates of change. The actual process was nothing new as I have done it before in my spare time, however I have a few questions about why and how we do it:

Is there a difference between instantaneous and average rates of change?
By 'instantaneous' we are referring at a specific point in time (or along the x-axis, depending on what you're differentiating, etc..). This is different from the 'average' because the average is exactly what it is, the total average change over the interval, it tells us nothing about the change at a specific point.

It is the difference between knowing what the exact velocity of a race car that is half-way through the track, and knowing its overall average velocity from start to finish. They are not the same.

What does "derivative" actually mean?
The instantaneous rate of change at some point, so like gradient at a specific point if you prefer.

Why do we only differentiate once (in GCSE)? Why not twice?
Dunno, it's usually above GCSE level.

Actually, why do we do it at all? I mean, I know the purpose is to find the rate at which y changes in respect to x (I think), but why can't we just substitute into the original f(x) instead of differentiating?
Substitute what into the original f(x)?

One of the purposes for differentiation is that we can consider a graph like and state what the gradient is at any point along this curve. Clearly, it is not a straight line so it doesn't have a constant gradient, it has a changing gradient as you move along it. By knowing the gradient at any point, we can in in fact construct the equations of the tangent lines to the curve at those points.

What does f'(x) mean? What does the apostrophe represent?
It means differentiated once with respect to . means has been differentiated twice over, and so on. After the third or fourth derivative we adopt the notation of to denote the nth derivative of
3. (Original post by irrelevant kid)
Is there a difference between instantaneous and average rates of change?
Yes. Think about speed (the rate of change of distance). You could walk for a bit, stop, run, sprint, then stop again. Your instantaneous speed (rate of change of distance) will constantly be varying, but the average speed (rate of change of distance) is just your total distance divided by the time.

(Original post by irrelevant kid)
What does "derivative" actually mean?
The rate of change of one variable with respect to another.

(Original post by irrelevant kid)
Why do we only differentiate once (in GCSE)? Why not twice? Actually, why do we do it at all? I mean, I know the purpose is to find the rate at which y changes in respect to x (I think), but why can't we just substitute into the original f(x) instead of differentiating?
You can differentiate a function as many times as you want, it just depends on what you're looking for. If you want stationary points, you want the first derivative (because you want to know when the rate of change is zero). If you want to characterise that stationary point, you want the second derivative (because you want to know the curvature of the function at that point). There are many applications in science and engineering where you need even higher order derivatives. Also, almost all functions can be expressed as an infinite series of derivatives.

I'm not entirely sure what you mean by your last question. Finding the derivative of the function is the only straightforward way of finding out the instantaneous rate of change of something with respect to another. Any other method is either going to be an approximation, or situational.

(Original post by irrelevant kid)
What does f'(x) mean? What does the apostrophe represent?
It's just notation. f'(x) means the first derivative of f(x) with respect to x, f''(x) means the second derivative of f(x) with respect to x, etc.
4. (Original post by RDKGames)
By 'instantaneous' we are referring at a specific point in time (or along the x-axis, depending on what you're differentiating, etc..). This is different from the 'average' because the average is exactly what it is, the total average change over the interval, it tells us nothing about the change at a specific point.

It is the difference between knowing what the exact velocity of a race car that is half-way through the track, and knowing its overall average velocity from start to finish. They are not the same.

The instantaneous rate of change at some point, so like gradient at a specific point if you prefer.

Dunno, it's usually above GCSE level.

Substitute what into the original f(x)?

One of the purposes for differentiation is that we can consider a graph like and state what the gradient is at any point along this curve. Clearly, it is not a straight line so it doesn't have a constant gradient, it has a changing gradient as you move along it. By knowing the gradient at any point, we can in in fact construct the equations of the tangent lines to the curve at those points.

It means differentiated once with respect to . means has been differentiated twice over, and so on. After the third or fourth derivative we adopt the notation of to denote the nth derivative of
(Original post by Plagioclase)
Yes. Think about speed (the rate of change of distance). You could walk for a bit, stop, run, sprint, then stop again. Your instantaneous speed (rate of change of distance) will constantly be varying, but the average speed (rate of change of distance) is just your total distance divided by the time.

The rate of change of one variable with respect to another.

You can differentiate a function as many times as you want, it just depends on what you're looking for. If you want stationary points, you want the first derivative (because you want to know when the rate of change is zero). If you want to characterise that stationary point, you want the second derivative (because you want to know the curvature of the function at that point). There are many applications in science and engineering where you need even higher order derivatives. Also, almost all functions can be expressed as an infinite series of derivatives.

I'm not entirely sure what you mean by your last question. Finding the derivative of the function is the only straightforward way of finding out the instantaneous rate of change of something with respect to another. Any other method is either going to be an approximation, or situational.

It's just notation. f'(x) means the first derivative of f(x) with respect to x, f''(x) means the second derivative of f(x) with respect to x, etc.
Thanks! I understand it quite a bit more now and it makes sense

The question which neither of you understood (sorry, my wording was weird and I lacked specificity): I was just wondering if the differentiated function of a curve is the same all the way round? Like, is it universal for all the points of the curve?
Sorry if I'm not really making sense.
5. (Original post by irrelevant kid)
Thanks! I understand it quite a bit more now and it makes sense

The question which neither of you understood (sorry, my wording was weird and I lacked specificity): I was just wondering if the differentiated function of a curve is the same all the way round? Like, is it universal for all the points of the curve?
Sorry if I'm not really making sense.
Yes the value of is true at any point on , there isn't another for it, if that's what you're asking.
6. (Original post by irrelevant kid)
Thanks! I understand it quite a bit more now and it makes sense

The question which neither of you understood (sorry, my wording was weird and I lacked specificity): I was just wondering if the differentiated function of a curve is the same all the way round? Like, is it universal for all the points of the curve?
Sorry if I'm not really making sense.
It's still difficult to understand what you mean
7. (Original post by RDKGames)
Yes the value of is true at any point on , there isn't another for it, if that's what you're asking.
Okay, thanks a lot! I appreciate it
8. (Original post by irrelevant kid)
I was just wondering if the differentiated function of a curve is the same all the way round? Like, is it universal for all the points of the curve?
Sorry if I'm not really making sense.
Yes, it's valid for the domain that the original function is valid for (i.e. if your function works with any value of x, then the differentiation function will also be valid for any value of x).

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