00:00Hi everyone. So our next
00:02rule for differentiating is the
00:04product rule and we use
00:05the product rule for guess
00:06when when we have a
00:08product and this is our
00:09u times v. So it's
00:11one function times another function.
00:13Here I have x squared
00:14times sine of x. So
00:17when do we know when
00:18to use a product rule?
00:20If we have a product
00:21of two functions and we're
00:22being asked to differentiate it.
00:24Note the difference between let's
00:26say
00:28Let's say sine of 3x
00:32squared, this is not the
00:34product rule, this is a
00:35function within a function, this
00:36would be the chain rule.
00:38The product rule is a
00:41function times a function, so
00:43x squared times sine of
00:45x, that's a common misconception
00:48and make sure you understand
00:50both the product rule and
00:52the chain rule. And then
00:53we're going to do the
00:53quotient rule after which is
00:55for when
00:56when we are dividing. So
00:58let's look at this. We
01:00need a u and we
01:01need a v. So at
01:03the side over to the
01:05right, I want you to
01:05write u equals one of
01:08them, let's say x squared.
01:09It actually doesn't matter for
01:11the product rule because u
01:13times v is the same
01:14as v times u, but
01:15what we may as well
01:16stick with the first one
01:16is u. And then the
01:18second one is sine of
01:20x. Don't skip this step.
01:22I see many people go
01:23straight to the
01:24straight to the formula, try
01:26and sub it in here
01:26without doing the calculations at
01:29the side and guess what
01:30it often needs to error.
01:32So take your time at
01:33the side, you got u
01:33equal the first one, v
01:34equals the second one, then
01:36you differentiate u, you get
01:38du dx equals 2x, then
01:41you get dv dx, dv
01:45dx equals cos x. Now
01:50you're ready to use this
01:52formula
01:52Now we say dy dx
01:55equals u times, you don't
01:58have to write out the
01:59formula every time, write it
02:01down here. U times dv
02:02dx plus v times du
02:06dx. What is u? u
02:09is x squared. dv dx
02:12is cos x, so it's
02:14x squared, cos x plus
02:17what is v sin x?
02:20times d u dx, which
02:24is 2x. Probably best to
02:29write it as x squared
02:31cos x plus 2x sin
02:36x, but that's it, done.
02:40Fairly straightforward, just recognize that
02:44it is product rule, and
02:46then take your time and
02:48apply
02:48I what you need to
02:50do to the formula. I
02:52want to do one more,
02:53one more example. So here
02:55we have f of x
02:57equals e to the 2x
02:59times 5x plus 3. At
03:01the side, what is u?
03:03u is e to the
03:052x. v is 5x plus
03:093. Now note, you can
03:11have a chain rule within
03:14a product rule.
03:16This is a chain rule.
03:17So this is d u
03:18dx equals e to the
03:222x times 2. And if
03:24you are okay with me
03:26doing this, I'm going to
03:27do 2e to the 2x.
03:29Just so I just want
03:30the 2 before. But it's
03:31e to the 2x times
03:33the derivative of 2x, which
03:34is 2. So it's e
03:35to the 2x times 2,
03:37which is 2e to the
03:382x. And d v dx
03:39is equal to 5.
03:44I'm going to write out
03:47the formula again just because
03:49you don't have it. dy
03:51dx is u times dv
03:54dx plus v times d
03:59dx. u is e to
04:02the 2x. dv dx is
04:055, so times 5. v
04:08is 5x plus 3. Make
04:11sure you put in a
04:11bracket, 5x plus
04:12was 3 and then du
04:14dx is 2e to the
04:172x. How well do you
04:22want to simplify this? Well
04:23look, I'm going to do
04:25something a little bit fancy.
04:28I'm going to take out
04:29the e to the 2x
04:30because there's an e to
04:31the 2x in this term
04:33and an e to the
04:342x in this term and
04:35I'm left with a 5
04:38here and I'm not
04:40with 2 times 5x plus
04:433 here, 2 times 5x
04:46plus 3, close the bracket,
04:51e to the 2x, e
04:55to the 2x times 5
04:59plus 10x because 2 times
05:035x is 10x plus 6
05:06and then finally
05:08e to the 2x into
05:1210x plus 11, it's 5
05:16plus 6, it's 11. Okay,
05:19that's the product rule. Look,
05:19I didn't have to bring
05:20it down to this form,
05:22but I liked it. Why
05:24not? It's most simplified form.
05:27So, a product rule, as
05:28I said, you're given it
05:30in the formula booklet. Make
05:32sure that it's a product,
05:33so it's something times something
05:35else. And then,
05:36You as one of them,
05:38via the other one, differentiate
05:39them subliment to the formula
05:40and simplify as appropriate. Next
05:44lesson is the quotient rule.