Next I’m going to explain the definition of derivative of a function in a point. It’s an abstract concept, but you need to know it.
The definition of derivative makes more sense in Physics. In Mathematics we are going to focus mainly on the calculation of derivatives, but it is necessary to know the definition in order to be aware of what we are calculating.
To understand the definition of derivative, you must understand two previous concepts that I will explain below: the mean variation rate and the instantaneous variation rate.
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Medium variation rate
There are cases in which we are interested in knowing how one variable varies with respect to the variation of the other variable.
For example, in Physics, for the function of speed, which relates the space traveled to time, it is interesting to know what the space traveled has been in a given period of time.
If time is the variable x and the space traveled is the variable y, then we are analyzing how the values of y vary as a function of the values of x.
As we already know, the value of a function at a point is the result of replacing x with a value in the function. Normally, for each value of x, the function has a different value. Therefore, the value of “y” varies with respect to the value of x
In general, we can calculate the variation of the value of a function in a certain interval of x, dividing the increment of the variable “y” by the increment of the variable x, knowing that the increment of a value is the subtraction of the final value minus the initial value, that is, dividing the subtraction of the value of the final function minus the value of the initial function, by the subtraction of the final x value minus the initial x value:
This is what we call the average rate of change.
The average rate of change is the average value taken by the “y”, with respect to the changes of x, in a closed interval.
If we represent it, we see that for the value x=a, the function takes the value f(x) and for the value x=b, the function takes the value f(b).
The variation or increase of the values of x is calculated as b-a and corresponds to the green horizontal segment.
The increment of the “y” values corresponds to the green vertical segment, calculated as f(b)-f(a).
Let’s see it with an example that surely is much clearer to you: Calculate the mean variation rate of the following function, in the closed interval [0,3]:
First we calculate the value of the function for x=0:
We calculate the value of the function for x=3
And now we apply the formula for the average rate of variation:
Whose result equals 4.
What does this 4 mean? That we can take as mean value, that in this function, every time the value of x is increased by one unit, the value of “y” is increased by 4 units.
Keep in mind that it is an average and approximate value. The average rate of variation only serves to give us an idea of what the values of “y” vary from those of x, but it does not serve to calculate the value of the function for x=2, for example.
Instantaneous variation rate
Let’s consider now that the value of x is increased by “h” units, that is, the initial value is x=a and the final value is x=a+h. (we consider the increment of “h” units to control how much x varies):
The average rate of change in this case would be:
You see, now the average rate of variation depends on h.
When we calculate the mean variation rate, as I mentioned before, we are calculating approximate mean values for a certain interval of x values, but the most normal thing is that within that interval, not all points have the same variation rate. In fact, the rate of variation does not vary only in linear functions.
If we consider h as a very small value, so small that it is very close to zero, we would be fine-tuning more and more in calculating what is the exact rate of variation at a given point, since in that case, the final value of x would be practically the same as the initial value and we would be in a very small range of x values.
If we take that value of h to the extreme, that is, that it tends to zero, we would obtain the value of the variation rate of a specific point. This concept is called the instantaneous variation rate and is calculated as the limit when h tends to zero of f(a+h)-f(a) divided by h. This concept is called the instantaneous variation rate and is calculated as the limit when h tends to zero of f(a+h)-f(a) divided by h:
For example, calculate the instantaneous variation rate of point x=2 in the next function:
We apply the instantaneous variation rate formula:
We operate on the numerator:
We take a factor common to h, so it is annulled in the numerator and denominator and we get the result of the limit:
As you can see, although the average variation rate calculated for this same function was 4 in the interval [0.3], where 2 is located, the instantaneous variation rate for x=2 is equal to 5.
An example of this difference between the mean value rate and the instantaneous value rate we have with velocity. For example, to go from point A to point B, a car can take 2 hours at an average speed of 100 km/h. That doesn’t mean his speed is 100 km/h during the 2 hours. On your way, you may find sections where there will be more traffic and your speed is 80 km / h and others where you can go faster and your speed is 120 km / h.
Definition of derivative of a function in a point
The derivative of a function in a point corresponds to the value of the instantaneous variation rate in that point and is therefore calculated as:
The derivative of a function in a point can be written in all these forms, all equally valid:
Let’s see an example of calculation of the derivative of a function in a point: Calculate the derivative of the following function in the point x=2:
This example is very similar to the previous one, since calculating the derivative of a function in a point is equivalent to calculating the instantaneous variation rate.
We apply the formula of the derivative of a function in a point:
We operate to subtract the fractions from the numerator and divide the result by h:
We remove the h from the numerator and denominator and solve the limit:
1- Calculate the mean variation rate of the interval [1.5] and the instantaneous variation rate of point x=3 of the following function:
2- Calculate the mean variation rate of the interval [3.5] and the instantaneous variation rate of point x=4 of the next function:
3- Calculate the derivative of the following function at point x=4:
4- Calculates the derivative of the next function at point x=1: