Transcription
If we have a function y equal to f of x, we now know how to find its derivative at a particular value of x. So, related to each value of x, we get a number which is equal to the slope of the tangent line there. Therefore, can you tell me if the derivative of a function is also a function? Yes, the derivative of a function gives us a new function.
In this video, we will understand what this means as well as its implications. For example, consider the function y is equal to x squared. Previously, we'd found that at a particular value of x, let's say x1, the derivative of this function is equal to 2 times x1. But now notice that x1 is an arbitrary value of the variable x. That is, it can be any value of x. So let's replace it by x here. And also, let's say the derivative is denoted by a variable y Prime. Therefore, this means that for each value of x there is the value y prime related to it which is the derivative of f of x at that point. It means this relation is a function. So we see that finding the derivative of a function gives us a new function. This function is called the derived function. Here the independent variable is x while the dependent variable is y Prime. At a particular value of x, the value of the derived function f prime of x tells us the derivative of f of x at that point.
Now, let me ask you a question. What will be the domain of the derived function? Do you remember what we mean by the domain of a function? It's the set of all the values of the independent variable x for which the function is defined. So, for which values of x will the derived function not be defined? Let's say at a particular value of x, say x2, this function is not defined. Then obviously, we cannot find the derivative of this function at x2. So it means that this derived function will not be defined at x2. So if set 'a' represents the domain of this function, then the derived function can only be defined for the values of x in set 'a'. But, now in one of our previous videos, we saw that the derivative does not exist even at some values of x in the domain of the function. It's because this limit does not exist. So this means that at the values of x in set 'a' where the derivatives does not exist, the derived function will not be defined. Therefore, the domain of the derived function is equal to all the values of x in the domain of this function, at which the derivative exists.
Let's take an example to understand this. Consider this function we saw previously. We've seen that for this function, even though the function is defined at x equal to zero, the derivative at x equal to zero does not exist. Finding the derivative at other values of x is simple. For x greater than zero, we will get the derivative to be constant at a value of 1. And for x less than 0, we will get the derivative to be constant at a value of negative 1. Its graph will look like this. So we see that for this derived function, x equal to 0 will not be in its domain. But now notice that f prime is a function. So can we differentiate this derived function too? What does this mean? Let's find this out in the next part. This tells us that the variable y prime is a function of the variable x. So it means that as the value of x changes, the value of y prime changes. So we can find the instantaneous rate of change of y Prime with respect to x, that is the derivative of f prime of x. Similar to this, it will be equal to the limit of the average rate of change of f prime of x. Now since this is a derivative of a derived function of f of x, it's called the second derivative of f of x. So it's denoted by putting double Prime on f like this.
Now, as seen earlier, the differentiation of f prime of x gives us a new function called the second derived function. So we can further differentiate it. This gives us the third derivative of f of X. Similarly, we can find the fourth derivative and so on... These are all called the higher-order derivatives of f of x. But what do these higher-order derivatives tell us? Do they have any significance? Well, as we look at advanced topics of calculus, we will understand why the higher-order derivatives are significant. For now, let's look at an example and understand the importance of the second derivative. Consider this object in motion. We note that the speed of the object tells us how fast or slow it changes its position. That is the rate of change in position with respect to time. But it's the speed of any object in motion always constant? No, right. For example, as the object starts from rest, its speed starts increasing from zero. We say that it accelerates. And when the object has to come to rest, its speed decreases and we say that it decelerates. So acceleration of an object is the rate of change in speed with respect to time. And we know that speed is the rate of change in position with respect to time. So we see that acceleration tells us the second-order change in the position of the object.
Let's see what we mean by it. Let's say the object's position is given by this function of time. We have seen previously that the instantaneous speed of the object at a time t1 will be equal to the derivative of this function at t1. So the derivative of this position function will give us a new function. This function will tell us the speed of the object at a time (t). Now, can you tell me how we can find the instantaneous acceleration at time t1? It will be given by the derivative of the speed function at t1, right? This means the acceleration function will be given by the second derivative of the position function. So we know that the second derivative of a position function tells us about one important physical quantity. But wait, there is something more fascinating. It turns out that for any object in motion, the force acting on it is equal to its mass times acceleration. We know that the mass of an object is constant. So we see that the second derivative of the position function tells us the force acting on the object. Therefore, while studying the motion of an object, finding the second derivative of the position function plays a central role.
So until now, we've seen many aspects of the derivative of a function. Also, we've seen related to this idea that there are many notations involved. One particularly important one we haven't talked about is this one: dy by DX or DF by DX. This notation suggests that the derivative of a function is a ratio of two quantities. But we know this is not the case, as we've seen before, the derivative is the result of the limit of this ratio. It's not a ratio in itself. But we will see how useful this notation is as we delve deeper into calculus. In essence, it captures the whole idea of the derivative; let's see how. Consider this function; this symbol Delta is always used to denote the change in a quantity; so if the value of x changes from x1 to x2, then Delta x means x2 minus x1. In other words, x2 will be equal to x1 plus Delta x. Now let's say at x1, the value of the function is y 1 and at x 2 the value of the function is y 2. So Delta y means y2 minus y1. Therefore, the numerator here can be denoted as Delta y. Now Delta x tends to 0 means the change in x will get very very small, close to 0. We say that the change is infinitesimal. We denote this infinitesimal change in x by dx. Now we see that as Delta x tends to 0, Delta y will also get smaller and smaller, so we say that the change in y is also infinitesimal and we denote it by dy. Now since the derivative is the limit of the ratio change in y over change in x, we can intuitively think of it as the ratio of infinitesimal change in y over infinitesimal change in x. It means the derivative tells us how the value of y changes for an infinitesimal change in X. Hence we can see that this notation helps us intuitively understand the derivative of a function. But always be aware that the derivative of a function is not a ratio of two quantities. It's a number which is a limit of the ratio of the two quantities. Would you rather have any other notations for the derivative of a function? Share your thoughts in the comment section below and subscribe to our channel to keep learning calculus.