Product Rule Example

00:01 So we've looked at the product rule in more than one way, we've looked at the product rule numerically, we've looked at the product rule and its definition.

00:11 And we've also done an exhaustive derivation of the product rule.

00:15 So let's do one more example before we move on to the next and the final rule.

00:22 Let's have a look at this questions y=x^3(3x^2-1)^4 First thing that we need to do is we need to spot what kind of function this is.

00:35 Firstly, you can see that we have a function x^3 and that is multiplying with another function.

00:41 So there's definitely a product rule.

00:43 But the second function, you will also notice is a function within a function.

00:47 So we're using more than one rule here to work out the differential.

00:52 Let's just remind ourselves of what the product rule states.

00:55 So the product rule says that dy/dx= vdu/dx+udv/dx or you could use the dash notation, whichever one you prefer.

01:08 It doesn't matter what order you put them in either because they're plussing, so you could put your udv/dx first, and then vdu/dx next because they're adding, it doesn't really make a difference.

01:20 So we're going to split this into two.

01:21 So we're going to say this is my u function, which is x^3 and this is my v function, which is (3x^2-1)^4.

01:31 We differentiate each one of them separately, so I have du/dx, which gives me 3x^2.

01:39 And then I have dv/dx, which gives me, remember that this is now the chain rule.

01:45 So firstly, the outside function, bring the power down 4, leave everything on the inside as it is decreased the power by 1, and then don't forget to multiply with the differential of the inside function, which is 6x.

02:00 This gives me a final derivative of 24x(3x^2-1)^3.

02:09 All of that is done.

02:11 So now it's just a matter of putting this into the product rule.

02:14 So the product rule is saying dy/dx, we want vdu/dx, so v is here.

02:20 So I've got (3x^2-1)^4.

02:25 I now need to multiply this with du/dx, which is this function.

02:29 So it's multiplying with 3x^2, put a plus in the middle because that's what the rule says.

02:34 We now take our u function, which is x^3.

02:38 And we multiply this with dv/dx, that's this function here.

02:43 So if I just put brackets around, it's because this has a bit more happening, I got (3x^2-1)^3.

02:51 And we just tidy this up, so I've got 3x^2(3x^2-1)^4.

03:01 So we multiply this term together, which gives me 24x^4, and then (3x^2-1)^3.

03:11 There's one last thing you can do here, you can see here that within this whole expression, you have a common factor of (3x^2-1)^3.

03:19 So you can take that out of this bracket.

03:23 This is just algebra now.

03:25 So just working on our algebra skills here, leaving us now.

03:28 So you've taken (3x^2-1)^3 out, so that leaves you with 3x^2(3x^2-1).

03:39 And in this term, here, you've taken the (3x^2-1)^3 out, leaving you with just 24x^4.

03:49 So this is just an extra bit of factorizing, if you would like to do, we can neaten this up, that gives you (3x^2-1)^3, multiply that through, we end up with 9x^4-3x^2+24x^4.

04:11 You can now see that you can add x to the 4x to the 4 terms together.

04:19 So got (3x^2-1)^3, you can add 9x^4 with 24x^4, giving you (33x^4-3x^2).

04:34 So just an extra little part of expanding things out and you can see that it's made the solution a little bit easier or a little bit easier to read than what we had here previously.

04:44 So we've just tidy this up to make it into a slightly more simpler, a bit more readable differential.

04:51 So we've now used the product rule to find the derivative for x^3 (3x^2-1)^4 We use the product rule and the chain rule, and we did some algebraic simplifying to get to our final answer.