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[MUSIC] What's the derivative of a 
product of two functions? The derivative 

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of a product is given by this, the 
Product Rule. 

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The derivative of f times g is the 
derivative of f times g plus f times the 

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derivative of g. 
It's a bunch of things to be warned about 

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here. 
This is the product of two functions, but 

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the derivative involves the sum of two 
different products. 

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It's the derivative of the first times 
the second plus the first times the 

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derivative of the second. 
Let's see an example of this rule in 

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action. 
For example, let's work out the 

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derivative of this product, the product 
of 1+2x and 1+x^2. 

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Alright, well here we go. This is a 
derivative of product, so by the Product 

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Rule, I'm going to differentiate the 
first thing, multiply by the second, and 

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add that to the first thing times the 
derivative of the second. 

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So, it's the derivative of the first term 
in the product times the second term in 

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the product, derivative of the first 
function times the second, plus the first 

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function, 1+2x, times the derivative of 
the second. 

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So, that's an instance of the Product 
Rule. 

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Now, this is the derivative of a sum, 
which is the sum of the derivatives. 

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So, it's the derivative of 1 plus the 
derivative of 2x times 1+x^2 plus 1+2x 

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times the derivative of a sum, which is 
the sum of the derivatives. 

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Now, the derivative of 1, that's a 
derivative of a constant function that's 

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just 0, 
this is the derivative of a constant 

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multiple so I can pull that constant 
multiple out of the derivative, 

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times 1+x^2+1+2x times, the derivative of 
1 is 0, 

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it's the derivative of a constant, plus 
the derivative of x^2 is 2x. 

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Alright. 
Now, I've got 0+2 times the derivative of 

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x. 
The derivative of x is just 1. 

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So, that's just 
2*1*(1+x^2)+(1+2x)*(0+2x). 

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So, there it is. 
I could maybe write this a little bit 

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more neatly. 
2*(1+x^2)+(1+2x)*2x. 

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This is the derivative of our original 
function (1+2x)*(1+x^2). 

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We din't really need the Product Rule to 
compute that derivative. 

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So, instead of using the Product Rule on 
this, I'm going to first multiply this 

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out and then do the differentiation. 
Here, watch. 

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So, this is the derivative but I'm going 
to multiply all this out, alright? So, 

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1+2x^3, which is what I get when I 
multiply 2x by x^2, plus x^2, which is 

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1*x^2+2x*1. 
So now, I could differentiate this 

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without using the Product Rule, right? 
This is the derivatives of big sum, 

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so it's the sum of the derivatives. 
The derivative of one, the derivative of 

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2x^3, the derivative of x^2, and the 
derivative of 2x. 

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Now, the derivative of 1, that's the 
derivative of a constant, 

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that's just 0. 
The derivative of this constant multiple 

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of x^3, I can pull out the constant 
multiple. 

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The derivative of x^2 is 2x and the 
derivative of 2-x, so I can pull out the 

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constant multiple. 
Now, what's 2 times the derivative of 

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x^3? That's 2 times, the derivative of 
x^3 is 3x^2+2x+2 times the derivative of 

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x, which is 2*1. 
And then, I could write this maybe a 

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little bit more nicely. 
This is 6x^2+2x+2. 

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So, this is the derivative of our 
original function. 

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Woah. What just happened? I'm trying to 
differentiate 1+2x*1+x^2. 

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When I just used the Product Rule, I got 
this, 2*(1+x^2)+(1+2x)*(2x). 

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When I expanded and then differentiated, 
I got this, 

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6x^2+2x+2. 
So, are these two answers the same? Yeah. 

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These two answers are the same. 
let's see how. 

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I can expand out this first answer. 
This is 2*1+2x^2+1*2x plus 2x*2x is 4x^2. 

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Now look, 
2, 2x^2+4x^2 gives me 6x^2. 

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And 1*2x gives me this 2x here. 
These are, in fact, the same. 

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Should we really be surprised by this? I 
mean, I did do these things in a 

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different order. 
So, in this first case, I differentiated 

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using the Product Rule and then I 
expanded what I got. 

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In the second case, first, I expanded and 
after doing expansion, then I 

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differentiated. 
More succintly in the first case, I 

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differentiated than expanded. In the 
second case, I expanded then I 

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differentiated. 
Look, you'd think the order would matter. 

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Usually, the order does matter. 
If you take a shower and then get 

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dressed, that's a totally different 
experience from getting dressed and then 

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stepping into the shower. 
The order usually does matter and you'd 

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think that differentiating and then 
expanding would do something really 

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different than expanding and then 
differentiating. 

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But you've got real choices when you do 
these derivative calculations, and yet 

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somehow, Mathematics is conspiring so 
that we can all agree on the derivative, 

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no matter what choices we might make on 
our way there. 

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And I think we can also all agree that 
that's pretty cool. 

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[MUSIC]