Hello and welcome to Excellence Academy. Today we'll be dealing on differentiation. What is differentiation? Differentiation has to do with the change in a parameter or variable with respect to another parameter or variable.
For instance, if I'm given the parameter, let's say y, expressed as a function of x, f of x, we can say that the change in y with respect to x is given by dy all over dx, which is equal to f prime of x. Also, if I'm given another parameter, let's say m, expressed as a function of t. Let's say f of t.
I will have that the change in m with respect to t is given by the m over the t. So it becomes the m over the t which is change in m with respect to t or perhaps it's also equal in value to f prime of t. So this is just the the very basic concept of differentiation. They change in a parameter with respect to another parameter. Having considered this, if I'm given a question to differentiate, what methods can I use?
What are different methods of differentiation? There are about six methods of differentiation. We'll pick each of them and deal with them one after the other.
The first method of differentiation is known as the general method some prefer to call it the par method number two is known as the first principle method that's my second method of differentiation number three is known as the chain rule called the chain rule perhaps also referred to as the function of a function or function of several functions right so this is read function of a function or functions of several functions right number four I have what is called the product rule Number five, I have quotient rule and one to this. Number six, I have implicit differentiation. right so these are like six different methods of differentiating tense you have general method you have first principle method you have chain rule also called function of a function or function of several functions but four you have the product rule number five you have the quotient rule number six you have implicit differentiation let's pick up the first one and explain this The first one I have there is called the general method. General method of differentiation. How does this work?
First up, let's assume I'm given a function y. i'm giving a variable per say y expressed as a function of x f of x and let's say this is now equal to x to a particular power or a particular index let's say n all right so i'm giving the function y being equal to f of x such that this is equal to x power a if i differentiate this now using general method i will have that the y over the x is equal to the idea is very simple i will multiply by the power i will subtract 1 from the power so from this now i'm having x is called the base and n is called the power or the index or the degree so multiply by the power here that's n so becomes n times i'm having what i have here um x power n subtract 1 from the part. Please observe.
I said multiply by the part. So n times this subtracts 1 from the part. n minus 1 such that if i work on this it's now equal to n times x gives you n x to power n minus 1 of course obviously n minus 1 so here's the little concept of general method of differentiation i'll pick an example and buttress this point let's say i have that y is equal to x to power 3 Right.
If I differentiate this, I'll have that dy all over dx is equal to, the power here is 3, so it becomes 3 times this. So I'm having 3 times what I have here, that's x to the power 3, we said subtract 1, becomes minus 1. This is now equal to, I'm having 3 times x is 3x into 3 minus 1. That gives you 2. So this is the simple answer. for this right so look at um we'll take other examples and look at much more cases on general method don't forget the concept is very simple multiply by the power and subtract one from the power let's take a second example example two let's assume i have a function um say y being equal to let's see x to the power 4 minus 6x cubed plus 2x squared minus 8x plus 11. All right let's say we have this y being equal to x to the power 4 minus 6x cubed plus 2x squared minus 8x plus 1. I'm asked to find the y with the x here. What do you do? From this I'll have that the y all over the x is equal to, I will use the same concept throughout all of this, right?
The concept is multiply by the power, subtract 1 from the power. From this here, the power here is 4. So it becomes 4 times x into subtract 1, 4 minus 1. Next up, I have a minus here, so it becomes minus. The power here is 3, so it becomes... three times what I have here. Six x into this.
Three minus one plus. Next up, the part here is two. So, it becomes what there?
Two times what you have here. Two x into what I have here. Two minus one.
Next up, minus. I'm having each x here. The part here is one. So, it becomes one times. 8x into what there?
1 minus 1. Finally, plus, I'm having a constant here. It becomes 0 times 11. Now, question, why is this a 0? Why do I have a 0 here?
You could say, I'm having a 0 here because there is no x attached to it. It's called a constant. For a constant, multiply the value there by 0. Why is this so?
I'll explain this. Let's say I have 11 here. Now, if you observe through this polynomial, you observe that I'm having x attached to each of the terms there.
So this is x to the power 4, x to the power 3, perhaps x cubed, x to the power 2, x squared. I'm having x here. So x is attached to all of these terms here, except 11. Now, if I have 11 standing alone, is there an x attached to it? Yes, we could.
express 11 in terms of x how if i have 11 we could say 11 is equal to what's there 11 times 1 so 11 is same as 11 times 1. i can express i can express 1 in terms of x using the zeroth law of indices from the zeroth law of indices we have that anything or any number raised to power 0 is 1 except zero all right so it's now equal to 11 1 in terms of x gives you what there x to power what 0 so x to power 0 is t1 so if i work on this now this is now equal to 11 x to power 0 hence 11 is the same thing as 11 x to power 0. so if i employ this method using general method we said multiply by the power the power here is 0 So it becomes 0 times 11. So that's how I got 0 times 11. All right. So next up, from this now, I'm now having equal to, I'm having 4 times x. That gives you 4x into 4 minus 1. That's 3. Minus 3 times 6 gives you 18x into 3 minus 1. That's 2. Plus, next up, 2 times 2 gives you 4x into 2 minus 1 is 1. That's 4x. Minus 1 times 8 is 8x into 1 minus 1. is 0 finally plus 0 times 11 obviously is 0 if i work on this now it's now equal to having 4 x cubed minus 18 x squared plus 4 x minus 8 into x plus 0 gives you what there 1 plus 0 it's off x plus 0 is 1 from the zed love indices n to the power 0 is what? 1. Alright, so this is now equal to 4x cubed minus 18x squared plus 4x minus 8 times 1 gives you what there?
Minus 8. So hence, this becomes the value for dy over dx. Looking at this, I can see like terms. These are even numbers. So I could factorize them using 2. so if i factorize this it's equal to 2 um it's as simple as dividing each of these terms by 2 so i'm having 4 divided by 2 gives you what's there 2 x to the power 3 minus here's 18 divided by 2 you have what there having 9 x squared plus here is 4 divided by 2 you have what there 2 so it gives you 2 x minus here is 8 divided by 2 you have order 4 so this becomes the factorized value for my dual over the x. Now for this now, I want to observe two things.
Number one is this, that if I differentiate a term such that it's just that term and x, at the end it gives me that term. So I'm having minus 8x here. If I differentiate minus 8x, it becomes just this term and x power 1. It gives you at the end minus 8. Next up, if I differentiate a constant, that means a term without x there, it gives me 0. so notice two things please if i differentiate a 10 such that it's just that 10 and x to power 1 at the end of the differentiation process it gives you that same term minus 8. if i differentiate a constant 11 having no x there it gives you 0. we pick up these two concepts um explain it better with an example let's say i have y being equal to 6x plus 7 for instance obviously from here I will have that dy all over dx is equal to notice here that it's just this term and x. So if I differentiate this gives you what here?
What I have here in that 6. So I'm having 6 plus this is a constant 7. Differentiate 7 you have what there? 0. So it becomes 0. So 6 plus 0 gives you what there? 6. This becomes my idea. So this is because my answer.
So this is the simple idea about general method of differentiation.