INTRODUCTION TO DERIVATIVES | CALCULUS

VIDEO TRANSCRIPT

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Hello and welcome to this video on derivatives. In this video I'll show you what a derivative is, its definition; we'll derive it mathematically, and prove that the derivative of x^n is n x^(n-1). Then I'll show you the derivatives of other elementary functions that pop up all the time and are important to know before finally ending by showing you the different ways that you might see a derivative written on the page. Let's start with the definition of what a derivative is. A derivative is defined as the rate of change of a function with respect to a variable. We can show this mathematically. If I get a set of axis... here's my x and y coordinate axis. If I draw a function of x, label this f(x)

and label a point on the function. To find the derivative, if we find the tangent at that point; this is dy/dx, that is the derivative of the function at that point. From looking at that you won't be able to see how mathematically you would write it down but I'll show you how you can. I'm going to label a second point that is a distance h away from first point. I'm going to label this point as (x + h). Now I'm going to draw a line that connects these two points, also known as the sectant line. We can write the equation of this line mathematically as (x + h); which is the second point; minus the first point over h. However, this isn't the tangent of point x, but as h gets smaller and smaller and smaller we're getting ever closer to the tangent of point x which is what we want. Mathematically we can write this as dy / dx or the derivative is as the limit of h goes to zero. So as it tends towards zero it will get closer and closer to the tangent line. This is described as the mathematical definition of a derivative. Knowing this, if our function is x^n we can use the mathematical derivative definition to find what the derivative of x^n would be and that's what we'll do right now. This is how to mathematically derive the derivation of x^n.

To begin, I'm going to rewrite out the definition. If the function is x^n that would look like this...using the binomial power series I'm going to write an expression for (x+h)^n ... and so on until you reach the end of the series. Where it's h^n, if we sub this expression into the definition above, we can make some cancellations. I'm going to take a factor of h out of the final terms just so you can see how the equation will be affected as h tends to zero. My factor of h; this is an optional step. You don't have to do this step but I'm including it just because it will make it easier to see what happens as h tends to zero. Everything inside the bracket here is just gonna be zero, therefore the differential for a function x^n is nx^(n-1). This is our first elementary derivative. Now I'll show you a few more elementary functions and their derivatives. The ones I'll show you are ones that you should know or that would be very useful to know because they are very common and they crop up all the time.

If our function is just a constant; denoted here by the letter a; then its derivative is just zero. If our function is in the form of x^n, its derivative would be nx^(n-1). The function ln x; its derivative is 1/x. And for the function e^x (I'm gonna put a constant there as well) then its derivative would be ae^(ax). If it was just e^x then its derivative is literally e^x. For the trig functions, all you need to remember; sine will differentiate to cos, cos will differentiate to -sine, -sine will differentiate to -cos and -cos will differentiate to sine. And it will carry on going and going. Drawing this circle will help immensely. I recommend that you remember the differentials of these elementary functions because they crop up a lot, especially in physics. Before I end the video, I'm going to show you the different ways that you can write down the derivatives.

In this video we've already seen a few different ways of writing down a derivative. We've seen dy/dx and we've seen f'x. There are a few other ways that we can write down or denote that something is a derivative or that you need to find its derivative. I'll begin with a function. You could also write this as y = and then it's the variable. There are a few ways that you might see the derivative expressed, so, as we've seen in the video before, the first derivative can be expressed as f'x but it could also be expressed as y' and this is the same as dy/dx. And it's very similar when it comes to the second derivative. Instead of f' it'll be f''. Same with y' it'll be y''. For the second derivative it's d^2y/dx^2. That just denotes that you've differentiated with respect to x twice. That's why y isn't squared but x is, it's because you've differentiated with respect to x. One thing that you might see; it's not common but I'll go through it just in case you do; sometimes you might see a number or a variable in the brackets. That just means you need to find the derivative of the function and then evaluate it at that value. So for this example; the first derivative find the first derivative and then evaluate it at 2. If the function was x^2 then the first derivative would be 2x so evaluating that at x = 2, it will be 2*2. So the first derivative of that evaluated at 2 would be 4. 

There's one more thing I'm going to mention before we wrap the video up and that's for when you are deriving with respect to time. It can have its own way of writing it. It's just a different way of writing it but it is exclusively only for time derivatives. If you do derive something with respect to time instead of y' it would be y-dot. Everything else will be the same. (it will) just be dt at the bottom for the fraction because it's with respect to time and for the second derivative instead of y'' it will be y-double-dot and everything else will be the same. Just swap the x for t's because it's with respect to time. 

So that about wraps this video for today. I hope it's been helpful and if you do have any questions feel free to post it in the comments below and I'll try and answer as quickly as I can. So with that thank you for watching and please subscribe if you found it helpful.