Unizor - Derivatives - Definition

Notes to a video lecture on http://www.unizor.com

Derivatives - Speed of Change

Consider a real function f(x)defined on a set of real numbers and taking real values.
It represents a dependency of some numerical value on certain input parameter. For example, temperature as a function of time or distance the car went as a function of amount of fuel spent, or the force of gravity as a function of an elevation above the surface of the Earth etc.

Let's assume further that this function f(x) is defined on some real interval (a,b) of its argument x, and we are analyzing the function's behavior around a particular point x₀ inside this interval. More precisely, we are interested to know how fast the function changes in the neighborhood of this point x₀∈(a,b).

Example 1

Back to one of our examples, consider the force of gravity on different levels of elevation. We can measure this force on any level and, comparing these forces G(h₀) and G(h₁) on two different levels h₀ and h₁, we can find the average speed of change of gravity on this interval per unit of elevation:
[G(h₁)−G(h₀)] ∕ (h₁−h₀)

Notice, this is an average rate of change of the gravity force on interval from h₀ to h₁, not the rate of change at point h₀ we are interested in, because there is a larger space between these two points for this rate to change.
What is also important is that the further points h₀ and h₁ are from each other - the further our average rate of change is from the one we are interested in. And, the closer point h₁ is to h₀- the closer our average rate of change of gravity is to the rate at point h₀ we need to know.

An obvious solution is to move point h₁ as close to h₀ as possible.
Mathematical solution is to know the function of gravity force as it depends on the elevation at any elevation level and calculate the limit of the average rate of change as point h₁ gets infinitely close to h₀:lim_h1→h0[G(h₁)−G(h₀)]/(h₁−h₀)

If this limit exists (and we should not assume it always exists for any function), it can be considered as a true rate of change at point h₀.

Example 2

Consider another practical problem
Assume, you are a policeman, who is measuring the speed of the passing cars in order to enforce some speed restrictions in the area.
Assume further that the only tool you have is the stopwatch.
To perform his job, policemen marks two points on the road - A₀ and A₁ - and, using the stopwatch, measures the time during which any car moves between these points.
Now the average speed on this interval from A₀ to A₁ will the length of this interval divided by time it took the car to cover it.
But what if the car moved faster than speed limit in the beginning of this interval and slower at the end, so the average speed was below maximum? To detect this, point A₁ should be as close to A₀ as possible.

Mathematical approach to this problem is as follows.
Let's start with some point on the road as an origin and mark each point with a distance from this origin.
Let's consider a function associated with the movement of a car - exact time t and distance from origin D(t) at each moment in time.
Point A₀ was reached at time t₀ and its distance from origin is D(t₀). Point A₁ was reached at time t₁ and its distance from origin is D(t₁).
The average speed on interval from A₀ to A₁ can be expressed as
[D(t₁)−D(t₀)] ∕ (t₁−t₀)

To get the speed at point A₀, we have to know the following limit:
lim_t1→t0[D(t₁)−D(t₀)]/(t₁−t₀)

Example 3

Assume, you are a reporter who has to report the news about the flood. What you need to report is something like this: "At time t₀ the level of water was rising with a speed of M meters per hour" (whatever the values of time t₀ required).

To accomplish this, you construct the function L(t) of the level of water at each moment in time. Considering you have the value of this function for all moments of time t, you can derive the speed of rising the water at any concrete moment t₀ using the following procedure.

You take the level of water at moment t₀, which is L(t₀), and at some moment t₁ after t₀, which is L(t₁).
The difference between these two levels of water signifies the rising of its level during a period from t₀ to t₁.
The ratio
[L(t₁)−L(t₀)] / (t₁−t₀)
is an average speed of rising water during this period.
If moment t₁ is closer to t₀, this average speed would represent the speed of rising the waters at moment t₀ more precisely, and the closer are these moments - the better would be the evaluation.

Exact speed of rising of the level of water at moment t₀ is, therefore
lim_t1→t0[L(t₁)−L(t₀)] / (t₁−t₀)

CONCLUSION

In all the above examples for a given function f(x) we were interested in the speed of its change at some specific value of the argument x₀.
This speed of change at x=x₀ is defined as
lim_x1→x0[f(x₁)−f(x₀)]/(x₁−x₀)

The limit above, if it exists (which might not be the case) is called a derivative of function f(x) at point x=x₀ and denoted as f'(x₀).

Since existing of this limit implies its identical value in any trajectory of x₁ approaching x₀, we can reformulate this using the following formula with identical meaning and notation Δx=x₁−x₀:
lim_Δx→0[f(x₀+Δx)−f(x₀)]/Δx
which is a more traditional definition of derivative of function f(x) at point x=x₀.