limit definition of derivative
When we've talked about derivatives so far,
we've talked about them in a very formal and time consuming way. And solve the limit definition of derivative
And What is the limit definition of the derivative? And limit definition of derivative examples
For instance, the instantaneous rate of change, we can think of the slope of the tangent line.
And one way to writethat is the slope of the tangent line is equal to the limit as x approaches a of f of x minus f of a divided by x minus a. Or,
we could write it as the limit as h approaches zero of fof a plus h minus f of a divided by h.
We could also call that f prime of a, the firstderivative at a.
We've also talked about the derivative as being a function.
So insteadof at a point a, we can say f prime of
x is equal to the limit as h approaches zero off of
x plus h minus f of
x all over h.
Well, if we had to do this every single time wefound a derivative, ...
What is the limit definition of the derivative
I am going to explain to you what a subsidiary is, the meaning of subordination with superpowers and how to track a subsidiary with definition.
And solve What is the limit definition of derivative
Also, I'm going to show you every bit of significant parts so you don't miss anything. Also, I know some of you seem to be in danger,
yet don’t stress that it’s not.
I guarantee.What is the limit definition of the derivative
Let me show you. So what's subordinate? The quick version does its job of revealing the soft line tendency for you to curve at any time. It gives you instantaneous speed of progress at any moment,
instantaneous speed of change at any point. Also,
we will sort out the valid meaning of the third derivative with the breaking point. In any case,
if you can barely wait to use it to calculate the derivative,
you can skip the skirting time and jump during filming. Or on the other hand you can godown the opening of this bani with us. Still,
say you have an arbitrary turn. Derivatives can line up at any time on your turning, steep or encouraged run-slant. But the thing it suggests to you is the slant of the lintigent until the turn.What is the limit definition of the derivative
Thus a degree of degradation is a line that intersects only one stage in the curve.
It skims the surface, bends and brushes contacts only at one stage.
There is an alternative slight degradation line at the end point of your chart.
So if the therapeutic reveals to you the tendency of every small degradation line,What is the limit definition of the derivative
Also read
how do you know that we know only one point on each degradation line without any doubt in this closed opportunity? Andy you realize we need two focuses to discover the trend.
Usually great investigation. We still don’t have the skills to do it. I mean, we'd rather have a linear function like a straight line chart, we could choose two focuses wherever we were, use the risk recipe, get a number for the slant and prevent it. What is the limit definition of the derivative
No.
You will end up, however, life is not so easy us our chart is crooked and the corner is evolving all over the place. The degradation line is unique everywhere.What is the limit definition of the derivative
So how might we know the incline all over if it's alwayschanging and we just know one point without a doubt on every digression line?
What is the limit definition of the derivative
Like I said, wedon't realize how to do that yet. We are not unreasonably complex yet. However,
we can figureit out from things that we do know and we can make the genuine definition or formulafor the subordinate.
Alright, so for the subsidiary on the off chance that we need to know the slant of thistangent line anytime call it x we can begin with what we do know the goodold slant of the straight line between two focuses. Pick a second point some place onthe bend and define a straight boundary through. What's more,
we realize how to discover the slopeof a straight line with the antiquated incline equation, which is a return tomiddle school presumably.
What is the limit definition of the derivative
So discovering the incline between two focuses has returned tohaunt you. However, you need to know your focuses. What is the limit definition of the derivative
So how about we mark them.
In the event that thissecond point is some distance away on a level plane, we should call that distance h,then this point has a x organize that is h more than x or x + h. We need theactual focuses the x, y sets. So for this point, for the x, y pair,
the x coordinatewill be x. Also, the y arrange we'll call f(x) which simply represents the yvalue that the capacity will give us. So that is that point. Furthermore, for this otherpoint, for the x, y pair,What is the limit definition of the derivative
the x arrange will be not x but rather x + h. Also, the ycoordinate will be not f(x), yet f(x) + h. Furthermore,
simply humor me. This is all goingsomewhere. We have two focuses, which is extraordinary, on the grounds that we can utilize those two pointsin the slant recipe that you know and love.
Alright, so here's the slant formulait's the ascent over run or vertical change over the level change. It's y2 - y1all over the x2 - x1. So we should utilize that for our two focuses. Alright,What is the limit definition of the derivative
so this is the incline wefound for the secant line. I did some rearranging and you could drop 'x's inthe base however this is the slant and it wound up being the distinction in the 'y'sover the even contrast, the h. So it's the slant of the secant line.
Andit's otherwise called the distinction remainder in the event that you hear that since it's a quotientof contrasts. Exceptionally unique name.
What is the limit definition of the derivative
This is the incline of the secant line.
What's more, weare most of the way to the subsidiary so hold on for me. So we are nearly to the derivative.
And we would be there. We would be done in the event that we had a straight-line chart. A linearfunction, at that point this slant we discovered would be correct all over. It would be enough.But we don't have that we have a bended chart. Also,
the straight line secant slopethat we discovered it's really an OK estimation for the incline at that pointx. It's a good guess. Furthermore,
that is a major piece of math is assessing something nonlinearwith something direct. So it's a good guess, the slant we found. Also,What is the limit definition of the derivative
it's notgreat however it's fair. In any case, we should not make due with good. We don't need anapproximation of the slant here. We need the specific incline, the genuine slant there andnot some wonky guess among it and some point close by. No,
we can make itexact on the off chance that we close in on x by narrowing h to nothing and picking a correct point that iscloser and nearer to one side so the flat distance gets more modest andsmaller.
Also, the nearer those two focuses are together the more precise ourestimate is for the slant by then. What's more, we can make it awesome on the off chance that we make h sosmall - make h endlessly more modest by adopting the breaking point as h strategies zero. So we'regoing to do a superior and better guess until it's right on the money.
So thesecant line turns into the digression line as this guide movements to x.
By the timeit's a similar point as x, this line is precise. It's the digression line slant. Sothe secant line turns into the digression line.
The incline of the secant line becomes theslope of the digression line. At the end of the day,
the restriction of the secant slant is thetangent line slant when it's just contacting at a certain point. Thus this breaking point is theslope of the digression line. Furthermore, shock, shock that limit likewise characterizes thederivative.
What is the limit definition of the derivative
This is the meaning of the subsidiary. So we have at last shown up atthe meaning of the subordinate. So this is the meaning of the subsidiary.
Whenthat limit exists, this characterizes that that is the subordinate. What's more, this notationyou just read as f'(x). It simply implies the subsidiary of f.
Additionally I should say becausethat slant,
digression line incline,
can be continually changing in better places - it's variable. It tends to be variable and that is the reason the subordinate is afunction.What is the limit definition of the derivative
It is itself a capacity very much like the first capacity additionally a function.Just to emphasize the subordinate discloses to you the incline all over the place - how steep, riseover run, slant, slant number, at the danger of seeming like a messed up recordderivative is essentially slant.
Furthermore,
to be straightforward this may not actually click andmake sense until you see it utilized in something like physical science where in the event that you'relooking at something moving on schedule,
the position changing on schedule, a bended graphwould imply that the slant is continually evolving.
The speed is changing and notconstant on the grounds that there's a speed increase.What is the limit definition of the derivative
While on the off chance that you had a straight-line graph,the incline would be the equivalent wherever in light of the fact that the speed isn't changingbecause there's no speed increase. In any case, this is the meaning of the subsidiary.
Presently the inquiry is.. how would you utilize it? Alright, so say you have tofind the subsidiary utilizing the meaning of the subordinate or by the cutoff process,this is the thing that you use. What is the limit definition of the derivative
You utilize the meaning of the subsidiary yet for yourf(x) - whatever you're given. So to track down the subsidiary f'(x) it's going to beequal as far as possible as h approaches zero.
That is there out front.
Breaking point as happroaches zero. And afterward we fill in this piece of the recipe. f(x) + h implies takeyour f, and instead of x, you utilize x + h, similar to the entirety of x + h instead of x everywherex shows up. So we should do that. The f(x + h) part... which resembles this3 times (x + h)^2, rather than x^2,What is the limit definition of the derivative
in addition to 12. At that point you deduct f(x), yet you'resubtracting all of f(x) similarly as is anyway the f(x) looks, however make sureyou use bracket when you do the deduction so you get the correct signs. Sowe have short all of f(x). And afterward it's all over h in the recipe.
Presently we can justdo some variable based math and rearrange. Alright, so here's everything to discover thederivative.What is the limit definition of the derivative
Our answer was 6x yet in the work, I thwarted. I appropriated. I factored.I dropped terms. At a certain point I figured out a h so it would drop with thebottom h. Also, in the end I took the cutoff.
What's more, I had the option to take the breaking point byplugging in zero for h. Furthermore, I got 6x which is a delightful, basic outcome after allthat work in variable based math. The subsidiary is simply 6x. f'(x) is 6x. What that implies isthat the subordinate of this capacity 3x^2 + 12 is simply 6x. Furthermore,What is the limit definition of the derivative
anyplace inthat work f, the incline would be the number you get by utilizing 6x at any instant,any x. The slant is the thing that you get from 6x right then and there. So that is the manner by which you findthe subordinate utilizing the definition with the cutoff. Simply recollect that... that is theonly precarious part is that this part f(x) + h implies that rather than x,
you utilize x + h. Soall of x + h instead of x. What's more, now, I should say, only FYI, as a publicservice declaration, this path with the definition and the breaking point is acceptable and all.It's right and illustrative,
yet in all actuality, it's really drawn-out. It's a lotof additional variable based math and by and by, that is not actually how a great many people takederivatives. So in case you're taking a great deal of subordinates, this additional polynomial math is,what's the word... a catastrophe. What is the limit definition of the derivative
Also, there's a quicker, easier way. What's more, if nothingexplicitly says you need to discover it utilizing the definition or by the cutoff process.
you can utilize the subordinate principles, which are a quicker, easier way. Substantially less extraalgebra. Furthermore,
Using the limit definition of derivative
I love math, but I don't love math this much.
So we are going to come up with some general rules that you'll need to memorize. And Using the limit definition of derivative
1. Let's first talk about aconstant
That is, where f of x equals a number c. And c is going to be a real number.
Remember, the derivative is the rate of change of the function. If I have a constant function, there is no rate of change. Therefore,
the rate of change is equal to zero.
This leads usto our first rule. The Constant Rule. If I have the derivative with respect to x of c,that equals zero if c is a real number.Using the limit definition of derivative
2. about is thePower Rule
The next rule we're going to talk about is thePower Rule. Before I prove it,
and I'm not going to do a lot of proofs, but I am goingto do this first proof.
But the Power Rule is if I take the derivative of a variablex to raised to the nth power,
that is equal to n times x to the n minus one power. And let's prove this one.
The first thing I am going to do is take my formal definition and start off with taking the derivative of x.
That is, x to the first power.
So if I let n equal1, that means f of x is equal to x to the first power, or just x.
And if I go aheadand use my definition of f prime of a, then I find that I have the limit as x approachesa of x minus a divided by x minus a which is simply equal to 1. Which,
by the way,
does match my power rule.
limit definition of derivative examples
Let's go ahead and check that. And this looks fine. The n is one, and I will talk limit definition of derivative examples
sothe derivative of x to the first power is 1 times x to the 1 minus 1, which is x tothe zero power. Anything to the zero power is 1, therefore my power rule does know that this would, in fact, equal 1. limit definition of derivative examples
So now let's say n is going to be greater than or equalto 2. A Andf of x is equal to x to the n power.
Now we can say the first derivative of f ofa is equal to the limit of x approaching a of x to the n minus a to the n, divided byx minus a.
limit definition of derivative examples
There is a factoring rule that we can always factor this x to the n minusa to the n into this. limit definition of derivative examples
So if I can factor x to the n minus a to the n in this form,
Isee that, first of all, I left off the limit.
Let me fix that.
There we go.
And now we see this x minus a divides out of the numerator and the denominator,
and I'm left with thefollowing. Once I have this in this form, I can go ahead and directly substitute x as a, because this is a polynomial (we remember our limit laws). If I go ahead and multiplythis all out,limit definition of derivative examples
I have a bunch of just a to the n minus ones. In fact, it's not just abunch to of a to the n minus ones,
there's exactly n of them. And so I get n time a to the nminus 1. limit definition of derivative
If I made this a function, I would find that f prime of x is equal to n timesx to the n minus one. Which is what I have for the power rule. This is the only one that I am going to go through and do a proof of, you don't have to recreate the proof,
butyou do have to be able to use the power rule. Notice that this power rule will actuallywork in the constants case.That is, if I had a constant,
limit definition of derivative.limit definition of derivative examples
say, c, it would be c times xto the zero. By the power rule, that would be zero times x to the zero minus one, butanything times zero is zero.
So the constant rule is really rolled into the power rule.By the power rule, the derivative of x with respect to x of x to the fifth, that's simplyequal to five times x to the five minus one, or,
5 times x to the fourth.
The second one - it would be really tempting to say the derivative with respect to x of 3 to the sixth is 6 times 3 to the fifth power,
limit definition of derivative examples.
but of course that's not right because 3 to the sixth is actuallya constant. There's no x in there. So this is still equal to zero. Don't fall in thattrap. Our next rule is going to be the Constant Multiple Rule.
That is, if I take the derivative of a constant c times a function f of x, that is simply equal to c times the derivativeof f of x.
I've got two examples up here.
limit definition of derivative examples
So my first one is, the derivative with respect to x of negative 5 sixths, x to the tenth power.
So the first thing I will do is pullout that constant.
limit definition of derivative examples.
So I have negative five sixths d dx x to the tenth. Now I am goingto use my power rule. And that gives me negative 5 sixths times ten times x to the ninth power. And if I simplify this,
I'll get the following. Negative 25 over 3 x to the ninth power.
Now my second example, I'm going to first again use the constant multiple rule to pull outthat 1 fifteenth. Notice now I'm taking the derivative of with respect to t. It worksthe same way.
Whatever my independent variable is. So, I've rewritten the square root oft as t to the one half power,
limit definition of derivative examples
because those are the same thing. And although I didn'tspecify, I'm going to go back and say with my power rule, that n has to just be a realnumber.
limit definition of derivative examples.
This I didn't prove, and Icould, but I'm just not going to. So I am going to saythat, using that same power rule, I'm going to get one over 15 times one half, that'smy exponent, times t to the one half minus one power.
Or, 1 over 15 times one half time t to the negative one half power.limit definition of derivative examples
I can rewrite this as such.
Generally, if I have startedoff with giving you information in square root form,
I am going to rewrite it as a square root and t to the negative one half power, that's just equal to one over the square root of t. And that's my final answer.
The next rule is the sum rule.
That is, the derivativewith respect to x of f of x plus g of x is simply equal to the derivatives of the separate functions added together.
I am warning you - do not assume that this is going to workwith products.
limit definition of derivative.
That is, with multiplication or division. But right now, we are just talking about addition and subtraction. And that is very straightforward.
Let's do a quick example. The derivative with respect to x of all of this is equal to the derivative of their individual terms. Again, this is by the sum rule.limit definition of derivative examples
When you're taking derivatives, you're not goingto have to write out each rule every time like we did with the limit rules, however,I want to specify this as I am teaching it so you can understand what I am doing,
step by step. The next rule I am going to use is the constant multiple rule.
And that allows me to pull the constants out. Finally, I am going to use my power and constant rules to come up with the following. And finally,
this is what I get as my answer. There's one more special derivative we're going to talk about.
And that's the derivative of e to the x. What is e? e is equal to an irrational number, 2 point 718 and a whole bunch of other digits.limit definition of derivative examples
limit definition of derivative examples
And what makes e special is this - that is, if I take e, and raise it to the h power,subtract 1 from it, and divide by h,
and let the limit of h approach zero, this is equalto the number 1. And here's a graph of e to the x, and the slope at zero is actually equal to one.
So where does this get us? So the derivative of e to the x is equal to the limitas h approaches 0 of e to the x plus h minus e to the x divided by h.
And again, that isby the definition of the derivative. So let's go ahead and rewrite my exponent as such.
I see both of my terms has an e to the x in them.
Well, I've already said that the limit of h approaching zero of e to the h minus 1 over h is equal to the number 1.
So that means that the derivative of e to the x is simply e to the x.
It's the best derivative out there.
And again,
I'll write it out. The derivative of e to the xis simply e to the x. One final thing to talk about -- I can take derivatives higher than just the first derivative. I could take, for instance,
the second derivative.limit definition of derivative examples
The second derivative of f of x is simply the derivative with respect to x of the first derivativeof f of x.
And I can expand that to talk about any values of n. And the one thing I want to point out is if I talk about the nth derivative,
I put that n in parentheses.
So that f to the parentheses n is not f to the nth power.
It's the nth derivative of f.
limit definition of derivative examples
So that justequals the derivative of the f to the n minus 1nth derivative of x.
And that's our first time of rules of differentiation.
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