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The measurement of the rate of change is a fundamental idea in differential calculus, which is concerned with the mathematics of change and infinitesimals. It enables us to identify the relationship amongst two altering variables and how they influence one another.

The measurement of the rate of change is also critical for machine learning like in application of gradient descent as the optimisation algorithm in training a neural network model.

In this guide, you will find out the rate of change as one of the critical concepts in calculus, and the criticality of measuring it.

After going through this guide, you will be aware of:

  • How the rate of change of linear and non-linear functions is measured
  • Why the measurement of the rate of change is a critical idea in differing domains.

Tutorial Summarization

This guide is subdivided into two portions, which are:

  • Rate of change
  • The criticality of measuring the rate of change

Rate of change

The rate of changes gives definition to the relationship of one altering variable with regard to another.

Take up a moving object that is displacing twice as much in the vertical direction, signified by y, as it is in the horizontal direction, signified by x. In mathematical terminology, this might be expressed as:

𝛿y = 2𝛿x

The Greek letter Delta 𝛿 is typically leveraged to signify difference or change. Therefore, the equation above gives definition to the relationship amongst the change in the x-position with regards to the change in the y-position of the moving object.

This alteration in the x and y-directions can be graphed by a straight line on an x-y coordinate system.

In this graphical representation of the object’s movement, the rate of change is indicated by the slope of the line, or its gradient. As the line can be observed to rise 2 units for every single unit that it runs to the right, then its rate of change, or its slope, is equivalent to two.

Rates and slopes possess a simple connection. The prior rate instances can be graphed on an x-y coordinate system, where every rate appears as a slope.

Connecting all of it together, we observe that:

Rate of change = 𝛿y / 𝛿x = rise / run = slope

If we had to take up two specific points, P1 = (2, 4) and P2 = (8, 16), on this straight line, we can go about confirming the slope to be equivalent to:

slope = 𝛿y / 𝛿x = (y2 – y1) / (x2 – x1) = (16 – 4) / (8 – 2) = 2

For this specific instance, the rate of change, indicated by the slope, is positive as the direction of the line is increasing in a rightward direction. But, the rate of change can even be negative if the direction of the line reduces, which implies that the value of y would be reducing as the value of x increases. Further, when the value of y stays constant as x increases, we would state that we possess zero rate of change. If, otherwise, the value of x stays constant as y increases, we would take up the rate of change to be limitless, as the slope of a vertical line is viewed as undefined.

 

Thus far, we have taken up the simplest instance of possessing a straight line, and therefore a linear function, with an unchanging slope. Nevertheless, not all functions are this simple, and if they were, there would be no requirement for calculus.

 

Calculus is the mathematics of change, so now is the best time to move on to parabolas, curves with modifying slopes.

 

Compared to the constant slope that personifies a straight line, we may observe how this parabola turns steeper and steeper as we shift rightwards.

Remember that the method of calculus enables us to undertake analysis of a curved shape by slicing it into several infinitesimal straight pieces arranged alongside each other. If we had to take up one of these pieces at some specific juncture, P, on the curved shape of the parabola, we observe that we identify ourselves calculating again the rate of change as the slope of a straight line. It is critical to keep in mind that the rate of change on a parabola is dependent on that specific point, P, that we happened to consider to begin with.

For instance, if we had to take up the straight line that goes through point P = (2,1), we identify that the rate of change at this point on the parabola is:

Rate of change = 𝛿y / 𝛿x = 1 / 1 = 1

 

If we had to take up a differing point on the same parabola, at P=(6,9), we identify that the rate of change at this point is equivalent to:

Rate of change = 𝛿y / 𝛿x = 3 / 1 = 3

The straight line that touches the curve at some specific point, P, is known as the tangent line, while the process of calculating the rate of change of a function is also referred to as identifying its derivative.

A derivative is merely a measure of how much one thing alters in contrast to another – and that’s a rate.

While we have taken up a simple parabola for this instance, we may likewise leverage calculus to undertake analysis of more complex non-linear functions. The idea of computing the instantaneous rate of change at differing tangential points on the curve stays the same.

We meet one such instance we come to train a neural network leveraging the gradient descent algorithm. As the optimization algorithm, gradient descent iteratively descends an error function towards its global minimum, every time updating the neural network weights to model better the training data. The error function is, usually, non-linear and can consist several local minima and saddle points. In order to identify its path downhill, the gradient descent algorithm computes the instantaneous slope at differing points on the error function, until it attains a point at which the error is lowest and the rate of change is zero.

The Criticality of Measuring the Rate of Change

We have, so far, taken up the rate of change per unit on the x-y coordinate system.

However, a rate can be anything per anything.

Within the context of training a neural network, for example, we have observed that the error gradient is computed as the alteration in error with regards to a particular weight in the neural network.

There are many differing domains in which the measurement of the rate of change is a critical concept too. A few instances are:

  • In physics, speed is computed as the modification in position per unit time
  • In signal digitisation, sampling rate is computed as the number of signal samples per second.
  • Within computing, bit rate is the number of bits the computer process per unit time
  • Within finance, exchange rates is a reference to the value of one currency with regards to another.

In either case, each rate is a derivative, and each derivative is a rate.