00:00Hi everybody. So in this
00:02lesson we are going to
00:03find the derivative using the
00:06power rule. So this is
00:07the power rule here. It's
00:08given in the formula booklet.
00:10And what it means is
00:12when you have a function
00:14f of x which equals
00:16x to the power of
00:17n such as some power,
00:19what we do is we
00:20multiply by the power and
00:22then we bring the power
00:23down by one. Okay, so
00:26that's sounds.
00:28pretty straightforward. Let me do
00:29some examples. So, why is
00:32this? How do we get
00:34the derivative? Well, the first
00:35thing is the notation is
00:38dy dx. If they give
00:41you f of x, the
00:42notation is f dash of
00:44x. This is light needs,
00:48remember, light needs a Newton
00:49both came up with calculus
00:50at the same time. Light
00:52needs to have this dy
00:53dx notation, and Newton had
00:56this f dash of x
00:58notation f dash of x.
01:00They mean the same thing.
01:02They both mean the derivative.
01:05So we're going to multiply
01:06by the power. What is
01:07the power? It's three. So
01:09we're going to do three
01:10times four. What's three times
01:12four? 12. And then we're
01:14going to bring the power
01:15down by one, which means
01:16subtract one. What's three minus
01:18one? Two. So that's the
01:20first term differentiated. Let's let's
01:24look at the formula booklet
01:27and related to this. So
01:29the power is n, so
01:30we multiply by the power
01:32n, so we multiply this
01:34by 3. And then we
01:36do n minus 1, which
01:37is subtract 1 from the
01:39power, which is 2. So
01:41I don't even look at
01:42this, I don't even look
01:44at this formula, and I
01:45don't ask my students to
01:47either just remember multiply by
01:50the power and bring the
01:51power down by 1.
01:52So 2 times 3, let's
01:53minus, so 2 times 3
01:55is 6, bring the power
01:56down by 1, x to
01:58the 1, or just leave
01:58it as x. Plus, now,
02:02this is like 7x is
02:047x to the power of
02:051. So I do 1
02:07times 7 is 7, and
02:09then I get x to
02:10the power of 0, but
02:11what's x to the power
02:13of 0? Because, I let
02:16me write it first, I
02:17do x to the power
02:18of 0, but x to
02:19the power of 0,
02:20is 1. So really it's
02:21just 7 times 1, which
02:23is 7. So the derivative
02:26of 7x is just 7.
02:28The derivative of 10x is
02:3010. The derivative of 20x
02:33is 20. That's actually quite
02:35an easy one to do.
02:36So it's just the derivative
02:38of negative 2x is negative
02:412. And then this one
02:43is easier even easier again
02:44because what? 5 is 5x
02:48to the 0.
02:48If you like, because x
02:51to the power of zero
02:52is one, and when we
02:54do zero times five, we
02:55just get zero. So the
02:57derivative of a constant is
02:59zero, and we don't even
03:00write anything. So if that
03:02was plus 52, we still
03:04write nothing, it's zero. Okay,
03:07next one. Multiply by the
03:09power. So as I say,
03:11we're doing f dash of
03:12x here, because this is
03:13Newton's preferred way of doing
03:14it. Four times a half,
03:16Hopefully you're okay with me
03:17writing two, four halves. If
03:19you four half, half of
03:22pizza, four half pizzas, you
03:24have two pizzas. So it's
03:26two x to the power
03:27of four minus one is
03:29three. Okay, now I've thrown
03:32in a k. So these
03:33five questions are all slightly
03:35different. So I've thrown in
03:36a k not to deliberately
03:38to confuse you, but just
03:39to get you comfortable with
03:41these things that can happen.
03:42So the k is a
03:43constant here.
03:44So this 5k x squared,
03:46so the 5k is behaving
03:47the same way this forwardage.
03:49So I'm not going to
03:50do anything with this, but
03:51I'm going to multiply it
03:52by 2. So 2 times
03:535k is 10k and I
03:56bring down the power by
03:571, x to the power
04:00of 1, which is just
04:01x. And then this minus
04:03x becomes minus 1, because
04:05remember, 7x becomes 7. 20x
04:07becomes 20, negative 1x, because
04:10there's a 1 there, negative
04:111x becomes
04:12negative 1 and the constant
04:15becomes 0. Here, 5 becomes
04:170. Here's 1, 1 becomes
04:190. 5. Next one. Okay,
04:23this is different again. I
04:25have an x squared below
04:27the line. So you cannot
04:29multiply by 2 and then
04:32bring the power down by
04:321. We've got a problem.
04:34When the x is underneath
04:35the line, you cannot differentiate
04:37it yet, but we can
04:39change it.
04:40So I'm not differentiating, I'm
04:43just changing. Hence, I'm going
04:44to write y again. I'm
04:46not writing dy dx, I'm
04:47writing y. So how do
04:48I change this? I need
04:49to get the power, I
04:52need to get that x
04:53above the line. So how
04:55do I bring an x
04:56above the line? Or what
04:57do I do to the
04:58power? Well, I can make
05:00that power negative two. Because
05:02if you remember from your
05:03laws of exponents, x to
05:05the power of negative two
05:06is 1 over x squared.
05:083x is about negative 2
05:10is 3 over x squared.
05:11So these are the same.
05:13This is the same as
05:14this. And then 5x is
05:15fine. I'll just leave 5x.
05:17But note, I haven't differentiated
05:19it yet. So I come
05:21on mistake. Well, certainly mistake
05:23I see too often is
05:25people change this into 3x
05:26to the minus 2 and
05:27then differentiate this into 5.
05:30But no, I haven't. I'm
05:32just changing this term. So
05:34this is the, this line
05:36is exactly
05:36the same as this line,
05:37just written differently. Now I
05:39can differentiate. So I'm gonna
05:41get dy dx equals, at
05:45this point, I can multiply
05:47by my power, negative six,
05:49x, subtract one for the
05:51power. What's minus two minus
05:53one? Well, it's minus three,
05:58be careful, it's not negative
06:00one. And then this five
06:02x, like my seven x
06:047 my 5x becomes 5.
06:08You can write this, this
06:09has negative 6 over x
06:11cube plus 5 if you
06:12want. He never told me
06:14to, so I'm gonna leave
06:15it like that. Next one.
06:17Again, I have an x
06:18underneath the line. What do
06:19I do? Well, firstly, I
06:20don't differentiate yet. I leave
06:23it as y. So y
06:25equals same thing. I'm gonna
06:27make this x to the
06:28minus 1. So like here
06:30I made it x to
06:31the minus 2. Here I
06:32can make it x to
06:32the minus
06:33is 1 plus, okay, I
06:37have x cubed over 3.
06:39You can think of this
06:40if you want as a
06:41third x cubed. So that
06:463, that third is gonna
06:48behave the same way as
06:50this 4 here. It's just,
06:52it's your coefficient, it's just
06:54a constant, it's just a
06:55third. So now I can
06:59differentiate dy dx
07:01Equals, negative one times k
07:04is negative k. x, what's
07:07minus one, or negative one
07:09minus one, is negative two.
07:12And then I have three
07:14times a third, what's three
07:15times a third? Well, three
07:17times a third is just
07:19one. So it's just gonna
07:20be one. I don't have
07:21to put the one obviously.
07:23So plus one, x squared,
07:26because I take one off
07:26the power. So that's it.
07:29negative k, x to the
07:30minus two plus x squared.
07:32Last one. Okay, now remember
07:35things don't always have to
07:36be x and y. It
07:38can be any, we can
07:39choose any letter we want.
07:40So I've just thrown in
07:41a p of t here,
07:42t, t is often used
07:43for time. So p of
07:45t is this, find the
07:47derivative. Okay, what's different about
07:51this question to the previous
07:52ones? There's a product, it's
07:55t multiplied by this.
07:57So what do we do?
07:58Well, we cannot differentiate this,
08:03differentiate this, undifferentiated this. It
08:05has to be, each term
08:08has to be in this
08:09form. You cannot have a
08:10product of two functions in
08:13what the function that I'm
08:15going to derive. So what
08:17I'm going to do is
08:18multiply out the product like
08:21this, t times 3t to
08:24the 5 is 3t.
08:25Three, all right, that property,
08:29three T to the six.
08:31Again, laws of exponents, T
08:33to the one times T
08:34to the five is T
08:35to the six. But I
08:36also have to do T
08:37times negative two, which is
08:39negative two, T. Now I'm
08:44able to differentiate P dash
08:47of T becomes multiplied by
08:51the power, six times three
08:52is 18.
08:53Bring the power down by
08:551, 2 to the 5,
08:56and then minus 2t becomes
08:58minus 2 again, like my
09:007x became 7, and my
09:035x became 5. Okay, hopefully
09:07that's clear. Multiply by the
09:09power, subtract 1 from the
09:12power. When you have an
09:13x underneath the line or
09:16a product like this, well,
09:18if you have an x
09:18underneath the line, you have
09:19to change it to a
09:20negative power.
09:21And if you have a
09:23product, multiply the product. Don't
09:24let the K's, if they
09:26throw in a K, don't
09:27let them confuse you. They're
09:29just there as, they just
09:32behave the same way your
09:34regular constants do.