Alright, in this video I'm going to do some examples involving taking derivatives using the good old chain rule.
Or as it's sometimes called. At least how I first thought about it, was being the outside-inside rule.
So, this is what the chain rule basically says:
It says you start off with a function that's actually a composition of functions.
And it says to take the derivative of that, which you do is
It's the derivative of "f", the inside is left alone, multiplied by the derivative of the inside.
So, in terms of this outside-inside rule
Notice you basically get the derivative of the outside function
Evaluated at the the inside function and then you multiply that by the derivative of the inside function.
That's the basic idea with the chain rule
And of all the derivative formulas that you'll run across
Probably the one that you'll use more often it's definitely going to be the chain rule.
It's one of these derivative rules.
It's a little confusing at first, but you definitely want to make that you get the handle on it.
Let's do an example here
So, suppose we have the function y=sin(x^2)
And we want to take the derivative the derivative of this.
Well, the outside function in this case is the sine function
and the derivative of sine is cosine.
If it was sine of "x" we would get cosine of "x," but instead of sine of "x" we have sine of "x^2"
So, the derivative is going to be cosine of "x^2"
and this is where we have to multiply by the derivative of the inside part. So, we'll multiply this by our "2x."
So, notice we've got,
if you think about it again the sine is being the outside, the "x^2" is being the inside
Notice we took the derivative of the outside part, we evaluated it at the inside part
and then we multiplied that by the derivative of the inside part.
So, that's a basic chain rule example.
Let's do some other ones here
Suppose I have the function
Let's say: y=x^2+3x+4
Suppose we had it evaluate with the third power
Well, if you had to you could always just multiply this out.
"x^3+3x+4" three times, combine your like terms and then can use your basic formulas, but-
suppose instead of to the third power I made it to the thirtieth power.
Probably not going to want to multiply this thing out thirty times and collect your like terms
You can now think about the outside function as being the stuff raised to the thirtieth power
and even kind of looks like it's the most outside.
So, if on the inside we just had "x" to the thirtieth, the 30 would come out front
So, the same thing happens here: the 30 would come out front. I'll leave the inside part alone.
I'll take one away, get to twenty-ninth power and then again I'll multiply this by the derivative of the inside stuff.
The derivative of "x^2" is "2x", the derivative of "3x" is just "3"
And there is your derivative.
I'm going to do some more examples of the chain rule on another video, some more complicated ones.
I might also have some more complicated chain rule problems on my website at JustMathTutoting.com
but, I'm also going to post more shortly here on YouTube, but-
I just wanted to do a couple quick basic examples to get you going.