Transcription
Derivative of a function measures its ‘RATE of change’. Imagine a quantity denoted by variable ‘Y’ which is continuously changing. But how it changes is CONTROLLED by another quantity, denoted by variable ‘X’. We can say that the variable ‘Y’, called the dependent variable, is a FUNCTION of the variable ‘X’, called the independent variable. The function ‘F’ tells us how the value of ‘Y’ changes with the value of ‘X’. The derivative of a function at a particular value of ‘X’ tells us the RATE of change of ‘Y’ with respect to ‘X’, at that particular value of ‘X’. That is how fast or slow the value of ‘Y’ changes with respect to ‘X’.
In our previous video, we saw how to find the derivative of a function. The whole process is summarised like this: It is called Differentiation. This ratio here is called the average rate of change between two values of ‘X’ which are ‘X not’ and ‘X not plus delta X’. If ‘delta X’ is greater than Zero, then ‘X not plus delta X’ will be somewhere here on the ‘X axis’. Then this ratio is the slope of this secant line between these two points. Now as we find the average rate in the interval closer and closer to ‘X not’, we see that these secant lines approach the tangent line at ‘X not’. This would be true even if we take ‘Delta X’ to be less than zero. Then ‘X not plus delta X’ will be somewhere here on the ‘X axis’. And as we find the average rate in the interval closer and closer to ‘X not’, we see that these secant lines approach the tangent line at ‘X not’. So we say that in the limit delta X tends to Zero, the average rate of change APPROACHES the instantaneous rate of change at X not. That is nothing but the derivative of the function at X not. We denote it by putting a dash like this on the notation for the function.
Note that the limit delta X tends to zero does not mean we put ‘Delta X’ equal to zero in this ratio. This would result in the ratio being equal to ‘zero divided by zero’, which does not make any sense. In this video, we will understand what this really means. Along with it, we will also explore different theoretical aspects of the derivative of a function.
Consider this simple example: ‘Y’ is equal to the square of ‘X’. Let’s say we want to find the derivative of this function at a particular value of ‘X’, say ‘X one’. Can you find this out? Actually, in one of our previous videos, we found the answer to this. We had found the instantaneous speed of an object for such a relationship between the distance travelled and the time elapsed. And to find the derivative we would first find the average rate of change. That is, how the value of the function changes as the value of ‘X’ changes by ‘delta X’. ‘F of X one plus delta X’ and ‘F of X one’ will be equal to this. After expanding this term and simplifying, we will get this. And after dividing by ‘delta X’ we will get this final expression. Now we know that to get the derivative, we will need to consider the interval ‘delta X’ tending to zero. So we put delta X equal to Zero here and get this. It is the derivative of the function at X one.
But now notice that all these expressions are equivalent to each other. So what if instead of putting ‘delta X equal to Zero’ here, we put ‘delta X equal to Zero’ here or here? We see that we will get the numerator and the denominator both equal to zero. But for this final expression, we got the answer as two times X one. So what is going on here? Why can we substitute ‘delta X equal to zero’ here and not here? Actually, substituting ‘Delta X’ equal to zero at any step here is not correct. Notice that in these steps we are dividing by delta X. And we know we can’t divide by ‘Zero’. What ‘delta X’ tends to zero means is that we are considering smaller and smaller values of ‘Delta X’. The values of ‘Delta X’ are close to Zero, but NEVER zero. But among all these steps here, delta X does not explicitly occur in the denominator here. So it is easy to see from this step that as delta X gets smaller and smaller, the average rate gets closer and closer to ‘2 times X one’. So substituting ‘Delta X’ equal zero here is just CONVENIENT to reach this conclusion. But always be aware of what it really means to put ‘Delta X’ equal to zero here.
Now notice one more thing. Although we had not explicitly stated it, here we took delta X to be greater than zero. That is, we are finding the average rate between X one and values of ‘X’ greater than it. But it is necessary to consider the case when delta X is less than zero. That is, finding the average rate between X one and values of ‘X’ lesser than it. As we find the average rate when delta X tends to zero, this average rate should approach ‘two times X one’. So let’s find the average rate in this case when delta X is less than zero. We can see the calculation for finding the average rate of change will be the same as this. But in the first case, this average rate will always be greater than ‘two times X one’. And in the second case, this average rate will always be less than ‘two times X one’. Now when delta X tends to zero, both these average rates will approach the same limit ‘two times X one’. Now we can conclude that the derivative of this function at X one is equal to ‘two times X one’.
So we see that to find the derivative of a function at a particular value of ‘X’, we have to find the average rate for two cases: When ‘delta X’ is greater than zero and when ‘delta X’ is less than zero. Now when delta x tends to zero, if both these average rates approach the same number, then that number is the derivative of the function at that particular value of ‘x’.
But consider this function now. It is called the absolute value or modulus function. These vertical bars say that the value of ‘Y’ is equal to only the non-negative value of X. That is, if ‘X’ is greater than or equal to zero, then ‘Y’ is equal to ‘X’. And if ‘X’ is less than zero, then ‘Y’ is equal to ‘negative of X’. Can you find the derivative of this function at ‘X’ equal to zero? Share your thoughts in the comments section below. We will find its derivative in the next lesson. Don’t forget to subscribe in order to get notified.