4. The Derivative as an Instantaneous Rate of Change

The derivative tells us the rate of change of one quantity compared to another at a particular instant or point (so we call it "instantaneous rate of change"). This concept has many applications in electricity, dynamics, economics, fluid flow, population modelling, queuing theory and so on.

Wherever a quantity is always changing in value, we can use calculus (differentiation and integration) to model its behaviour.

In this section, we will be talking about events at certain times, so we will be using Δt instead of the Δx that we saw in the last section Derivative from First Principles.

Note: This section is part of the introduction to differentiation. We learn some (much easier) rules for differentiating in the next section, Derivatives of Polynomials.

Velocity

We learned before that velocity is distance divided by time. But this only works if the velocity is constant. We need a new method if the velocity is changing all the time.

If we have an expression for s (displacement) in terms of t (time), then the velocity at any particular instant t is given by:

`v=lim_(Deltat->0)(Deltas)/(Deltat`

To make the algebra simple, we will use h for Δt and write:

`v=lim_(h->0)(f(t+h)-f(t))/h`

Continues below

Example

An object falling from rest has displacement s in cm given by s = 490t2, where t is in seconds (s).

What is the velocity when t = 10 s?

Note:

In the time given above, t = 10 s, the "s" (non-italic) is the official metric symbol for "seconds". Don't confuse it with s (using italics), which is the variable commonly used for displacement (as used in the first sentence of this Example, s = 490t2).

Answer

`s=f(t)=490t^2`

Here is the graph of s (displacement) against time (in seconds) . We see that the velocity (equivalent to the slope of the tangent of the curve) is not constant. At the beginning, the slope is `0` (the curve is horizontal, or momentarily flat). As time goes on, the object speeds up, and the slope of the curve gets steeper (more vertical), as we can see in the light grey tangent lines.

Now the velocity is given by:

`v=lim_(h->0)(f(t+h)-f(t))/h`

So we have:

`v=lim_(h->0)(f(t+h)-f(t))/h`

`=lim_(h->0)([490(t+h)^2]-[490t^2])/h`

`=lim_(h->0)([490(t^2+2ht+h^2)]-[490t^2])/h`

`=lim_(h->0)([490(2ht+h^2)])/h`

`=lim_(h->0)[490{2t+h}]`

`=980t`

So `v = 980t` is the expression that tells us what the velocity is at any time (`t ≥ 0`).

When `t= 10` s, `v = 980(10) = 9800\ "cm/s"`.

So the velocity at `t= 10` s is `98\ "m/s"`.

We write velocity as: `v=(ds)/dt` OR we can also write this as: `v=s`'.

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The derivative tells us:

  • the rate of change of one quantity compared to another
  • the slope of a tangent to a curve at any point
  • the velocity if we know the expression s, for displacement: `v=(ds)/(dt)`
  • the acceleration if we know the expression v, for velocity: `a=(dv)/(dt)`

Reader Question

A reader recently asked:

"Yes, but what does `dy/dt` really mean?"

Here was my reply:

In summary, `dy/dt` means "change in y compared to change in t at a precise value of t."

It is used where the quantity "y" is undergoing constant change. Let's use the example of temperature. Say you are in Melbourne, Australia (where daily extremes of temperature are common :-), and we want to know how fast the temperature is increasing right now.

In winter, at night, the temperature might typically be `2°"C"`. In summer (6 months later) at night, it may be `26°"C"`. The average rate of change is

`(26 - 2)/6 = 24/6 = 4^@` per month

This is a long term average change. It is not `dy/dt`.

But now let's think of one day in summer. At 6:00 am the temperature might be `13°` and by 1:00 pm it is (say) `27^@`. The average change now is

`(27 - 13)/7 = 14/7 = 2^@` per hour.

We still do not have `dy/dt`.

Now let's consider at 9:00 am it is `20°` and at 10:00 am it is `22.4°`. So the average change is

`(22.4 - 20)/60 = 2.4/60 = 0.04^@` per min (equivalent to `2.4°` per hour)

We could keep going for smaller and smaller time intervals (like second, then millisecond, then nanosecond and so on) to get a precise change in temperature at 9:00 am. This precise change is represented by the concept of `dy/dt`.

Historically, what I have described in The Slope of a Curve (Numerical) was what they had to do before Newton and Leibniz gave us differential calculus. In Derivative from First Principles we saw the algebraic approach that Newton and Leibniz developed. Now we can find precise values of `dy/dt` using a mathematical process based on a function, without having to substitute numbers all over the place.

Coming up...

In the next section, we will see some (much simpler) rules for differentiation. We won't use "differentiation from first principles" very often from here on, but it is good to have an understanding of where differentiation comes from and what it can do for us.