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

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

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

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

Â here. This is the product of two functions, but

Â the derivative involves the sum of two different products.

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

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

Â action. For example, let's work out the

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

Â 0:47

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

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

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

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

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

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

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

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

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

Â 1:37

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

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

Â just 0, this is the derivative of a constant

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

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

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

Â 2:50

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

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

Â out and then do the differentiation. Here, watch.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â So, this is the derivative of our original function.

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

Â 4:39

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

Â When I expanded and then differentiated, I got this,

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

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

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

Â 5:11

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

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

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

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

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

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

Â differentiated. More succintly in the first case, I

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

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

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

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

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

Â think that differentiating and then expanding would do something really

Â different than expanding and then differentiating.

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

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

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

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

Â [MUSIC]

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