00:00Hi everybody. So the first
00:03method we have for differentiating
00:05to find the derivative is
00:07the power rule. Now the
00:08power rule is given in
00:09the formula booklet. Here it
00:11is, section 5 .3. And
00:13we use it for functions
00:16of the form x to
00:19the n. And functions as
00:21the form x to the
00:22n, the derivative, the way
00:25we get the derivative is
00:26we multiply by the power
00:28So whatever this power is,
00:29we multiply in front by
00:30that. So it's n times.
00:32And then we bring the
00:33power down by one. So
00:34we subtract one from the
00:36power. Very straightforward. Like this
00:40example is going to be
00:42very easy for you. But
00:43they do get more difficult.
00:47And the reason they are
00:48difficult is not necessarily anything
00:49to do with differentiation or
00:52calculus. It's just the fractions,
00:55the roots,
00:56and the laws of exponents
00:58and just subtracting one from
01:01a fraction often causes problems.
01:03But certainly the main thing
01:05or the only thing really
01:06I want you to learn,
01:07get from this lesson is
01:09how to apply this rule.
01:12So firstly, are we able
01:14to apply the rule? In
01:15this case, yes, because we've
01:17x to the n, x
01:18to the n, that's like
01:20x to the one, and
01:21this is like x to
01:22the n because it's kind
01:23of like
01:24x to the zero if
01:25you like, x to the
01:26zero is 1. So let
01:28me just remove that though.
01:30Okay, so I'm able to
01:31differentiate. So when I differentiate,
01:34very straightforward. Multiply in front
01:37by the power. So I
01:38multiply by 3. And then
01:40subtract 1 from the power
01:423 minus 1 is 2.
01:45Then minus 2 times 2
01:48multiplied by the 4. Sorry,
01:50multiply by the power is
01:514.
01:52x to the power of
01:551, which is 2 minus
01:561 is 1. Plus, 5.
02:01Now, if this is x
02:02to the power of 1,
02:03it's 1 times 5, x
02:06to the power of 0,
02:07because 1 minus 1 is
02:080. But what's x to
02:10the power of 0? Well,
02:11it's 1. So we're actually
02:13just left with 5. And
02:15that's the reason why the
02:17derivative of
02:202x is 2, the derivative
02:22of 5x is 5, the
02:23derivative of 100x is 100.
02:26And then for constants like
02:287, imagine this to be
02:30x to the power of
02:310, well then I do
02:320 times 7 is 0
02:35and then it doesn't matter
02:35what comes after it because
02:37it's negative 1. What's 0
02:42times x to the power
02:44of negative 1, what is
02:45just 0? So the derivative
02:47of a constant
02:48is zero, derivative of 10
02:52is zero. And we're left
02:53with this. Obviously, you don't
02:56have to put in the
02:58one there. So it's just
03:013x squared minus 4x plus
03:045. And interestingly, well, because
03:07of the power rule, for
03:08a polynomial, the derivative of
03:10a polynomial, the degree will
03:12always be one less. So
03:14if this is a cubic,
03:15the derivative of a cubic
03:16is a quadratic. The derivative
03:18of quadratic is a linear.
03:21The derivative of a linear
03:23function is just a constant
03:26and the derivative of a
03:27constant is actually zero. Okay,
03:30next one. Now note, I
03:32can do y and dy
03:33dx and exactly the same
03:37I can do f of
03:38x and f dash of
03:39x. So am I ready
03:41to differentiate? Yes, because it's
03:42x to the power, x
03:43to the power, x to
03:44the power,
03:44So the derivative here is
03:47negative 4 times 2 is
03:49negative 8. x to the
03:51minus 4 minus 1 is
03:52minus 5. Fine. Plus 4
03:57times 3 over 2. Let
03:58me write the whole thing
03:59down. 4 times 3 over
04:012. x to the power
04:03of 3 over 2 minus
04:041 is 1 over 2.
04:08And then I'm going to
04:09have 3 times
04:12x to the x squared
04:15because 3 minus 1 is
04:17squared or is 2. And
04:19I have a 6 underneath
04:21the line. So I'll even
04:23like that. Now, yes, I
04:25do want you to simplify
04:26this. I can leave this
04:27as negative 8 x to
04:30the negative 5 or put
04:31it over x to the
04:325, whichever you prefer. These
04:35cancel four times 3 over
04:362 is, or you can
04:38say it's 12 over 2
04:39which is 6, so that's
04:40plus
04:406 x to the half
04:42and then this is obviously
04:443 over 6 is a
04:45half so let's just plus
04:47x squared over 2. Let's
04:50do it. Okay so those
04:53two I could differentiate right
04:56off the bat. Great. This
04:57one I'm not able to
04:59differentiate because it's not x
05:00to the power of something.
05:02The x is underneath the
05:03line is part of the
05:04denominator which is a problem
05:06and this x is in
05:07a root so it's not
05:08next to the power of
05:09something. So I'm not ready
05:11to differentiate. Now this is
05:12important. If we're not ready
05:14to differentiate, what we're going
05:16to do is we're going
05:16to change it so that
05:18we can differentiate, but we're
05:19not finding the derivative. I'm
05:21just going to write y
05:22equals because I'm just going
05:23to change the form of
05:25these. So instead of doing
05:27five over x, I'm going
05:28to do five x to
05:30the negative one. And I
05:32can do that. That's just
05:35one of the laws of
05:36to the negative one. And
05:39I can change this into
05:41x to the half, because
05:43the square root is to
05:44the power of half. Now
05:45I'm ready to differentiate now.
05:47It's a bit like this
05:48one over here. In fact,
05:48it's easier. So I'll get
05:50dy dx equals negative five
05:54x to the minus two,
05:56because I do minus one
05:57times five. And then I
05:58subtract one from the power.
06:00And then I do plus
06:01a half x, what's one
06:04half minus one
06:04minus one, well it's negative
06:06a half. Okay, done. I
06:12could change the way this
06:13looks if I wanted. I
06:15can make that one over
06:16two root x or negative
06:18five over x squared, but
06:20there's no reason to do
06:20it, so I'll leave it
06:21like that. Next one, am
06:24I ready to differentiate? No,
06:27because again, I've an x
06:28underneath the line, I've an
06:29x in a root here.
06:31There's a k there, which
06:33It's just gonna stay there
06:34don't let it make Don't
06:36let it scare you or
06:37make you think that this
06:39is much harder than this
06:40the K is just gonna
06:42act like a constant so
06:43I'm not ready to differentiate
06:44yet now a Common mistake
06:48is the mistake. I've just
06:50done there You actually write
06:52down f dash of x
06:53because you're so used to
06:54just as I was you're
06:56so used to writing it
06:57down that you just write
06:58it down, but I'm not
06:59differentiating. I'm just I'm
07:01I'm just getting f of
07:02x again, I'm just rewriting
07:03this. So this is gonna
07:05become 2k x to the
07:07negative 3. I want that
07:09x above the line, so
07:10I make it negative 3,
07:11like I made this x
07:12to the negative 1. And
07:15this becomes 5x to the
07:18power of now. If you
07:20remember my flower power rule
07:22that I love so much,
07:24my root and my flower,
07:29my root
07:29Oot is 3 and my
07:31flower power is 4. So
07:34it's x to the power
07:37of 4 over 3. Now
07:42I'm ready to differentiate. So
07:44I'm going to do f
07:45dash of x. Negative 3
07:48times 2k is negative 6k.
07:51x to the negative 4
07:53plus 5 times 4 over
07:563.
07:57Just do it in one
07:57go, it's 5, 10, 4
07:59is 20 over 3. So
08:01it's 20 over 3x to
08:04the 4 over 3 minus
08:061 is 4 over 3
08:09minus 3 over 3 equals
08:121 over 3. So if
08:14you 4 thirds minus 1
08:16is 1, 3rd. And that
08:20is finished. OK, I want
08:23to do one more question.
08:25when it is of this
08:27form. Now, I'm going to
08:29teach you something later that
08:31is called the product rule
08:32where you can differentiate products.
08:35Because here I have a
08:36product of two functions, x
08:38squared times this thing in
08:40the brackets. Now, the product
08:41rule is fine, you can
08:42use it, but at the
08:45moment, only knowing the power
08:47rule, what can I do?
08:49Well, I can multiply it
08:51out. If I multiply out
08:52the brackets,
08:53I'll get the function in
08:56the form that I want
08:58in order to use the
08:59power rule. So x squared
09:00times 3 over x is
09:02actually, well, let's write it
09:04all out. It's 3x squared
09:06over x plus 2x squared
09:10times x squared is 2x
09:12to the 4. Again, y
09:16equals 3x squared over x.
09:18One of the x's here
09:19is cancels with this one.
09:20I'm going to have a
09:213x
09:21plus 2x to the power
09:25of 4. Now I'm ready
09:26to differentiate dy dx equals
09:30the derivative of 3x is
09:313 plus 4 times 2
09:33is 8x cubed and that
09:38is it. Actually, the way
09:42I did it is easier
09:44than the product rule. So
09:47sometimes you have to decide
09:48what to use
09:49when to use the product
09:50rule or not. But if
09:51you can multiply it out
09:52and use the power rule,
09:53then why not just do
09:54that? Okay, so that's the
09:55lesson. The power rule, it's
09:59the first and main most
10:03popular way to find the
10:04derivative of a function. It's
10:06for functions in the form
10:07x to the n. And
10:09what we do is we
10:10multiply by the power and
10:12we subtract one from the
10:13power. Remember, do not differentiate
10:16until
10:17You are ready to do
10:18so like this example here.
10:20You are not ready to
10:21differentiate yet. And only write
10:23down dy dx when you
10:25are differentiating.