limit definition of derivative

limit definition of derivative

what is the derivative?

The Limit Of  derivative of function f at x=c is the limit of the slope of the secant line from x=c to x=c+h as h approaches 0. Symbolically, this is the limit of [f(c)-f(c+h)]/h as h→0. 
(x₁ – x₂)/(y₁ – y₂)
= (-1(-x₁ + x₂))/(-1(-y₁ + y₂))
= (x₂ – x₁)/(y₂ – y₁)

1. Write the expression replacing every x with (x + Δx).
(x + Δx)^2 + 5
= x^2 + 2xΔx + (Δx)^2 + 5

2. Subtract the expression, which gives Δy.
[x^2 + 2xΔx + (Δx)^2 + 5] – [x^2 + 5]
= x^2 – x^2 + 2xΔx + (Δx)^2 + 5 – 5
= 2xΔx + (Δx)^2

3. Divide by Δx, which gives the slope (Δy/Δx).
[2xΔx + (Δx)^2] / Δx
= 2x + Δx

4. Take the limit as Δx approaches 0.
lim Δx→0 2x + Δx
= 2x

This procedure is rarely done, as there are many shortcuts.

The solution to your example, f(x) = 6x^4 – 7x^3 + 7x^2 + 3x – 4:

6(x + Δx)^4 – 7(x + Δx)^3 + 7(x + Δx)^2 + 3(x + Δx) – 4
= {6[x^4 + 4x^3Δx + 6x^2(Δx)^2 + 4x(Δx)^3 + (Δx)^4]
– 7[x^3 + 3x^2Δx + 3x(Δx)^2 + (Δx)^3]
+ 7[x^2 + 2xΔx + (Δx)^2]
+ 3x + 3Δx
– 4}
– {6x^4 – 7x^3 + 7x^2 + 3x – 4}

= 6x^4 – 6x^4 + 24x^3Δx + 36x^2(Δx)^2 + 24x(Δx)^3 + 6(Δx)^4
– 7x^3 + 7x^3 – 21x^2Δx – 21x(Δx)^2 – 7(Δx)^3
+ 7x^2 – 7x^2 + 14xΔx + 7(Δx)^2
+ 3x – 3x + 3Δx – 4 + 4

= [24x^3Δx + 36x^2(Δx)^2 + 24x(Δx)^3 + 6(Δx)^4
– 21x^2Δx – 21x(Δx)^2 – 7(Δx)^3
+ 14xΔx + 7(Δx)^2
+ 3Δx]
/ [Δx]

= 24x^3 + 36x^2Δx + 24x(Δx)^2 + 6(Δx)^3
– 21x^2 – 21xΔx – 7(Δx)^2
+ 14x + 7Δx
+ 3

LimΔx→0 [24x^3 + 36x^2Δx + 24x(Δx)^2 + 6(Δx)^3
– 21x^2 – 21xΔx – 7(Δx)^2 + 14x + 7Δx + 3]
= 24x^3 – 21x^2 + 14x + 3

limit definition of derivative

limit definition of derivative,Long story short it’sa function that tells you the slope of the line tan gent to the curve at any point. Itgives you the rate of change (limit definition of derivative)at any instant, the instantaneous rate of changeat any point. And we’re going to figure out the proper definition of thederivative with the limit. But if you just can’t wait to use it to calculate thederivative, you can skip ahead and jump to the time in the description. Or you can godown this rabbit hole with us. But say you have a random curve.

 

The derivativetells you the slope at any point on your curve, the steepness or riseover run – the slope. But really what it’s telling you is the slope of the (limit definition of derivative)  linetangent to the curve. So a tangent line is a line that touches at only one point on thecurve. It skims the surface, grazes the curve and touches at only one point. Andevery point on your graph has a different little tangent line.

So if thederivative’s supposed to tell you the slope at each little tangent line, how canwe know that slope if we only know one point for sure on each tangent line? (limit definition of derivative) Andyou know we need two points to find the slope. Very good question. We don’t knowhow to do that yet. I mean if we had instead a straight-line graph a linearfunction like that, we would know the slope everywhere because you could pickany two points there, use the slope formula, get a number for slope and itwould be the same everywhere that slope.

limit definition of derivative

You’d be done. But life is not that simpleusually. Our graph is curved and the slope is changing everywhere. The tangent lineis different everywhere. So how can we know the slope everywhere if it’s alwayschanging and we only know one point for sure on each tangent line? Like I said, wedon’t know how to do that yet. We are not that sophisticated yet. But we can figureit out from things that we do know and we can make the actual definition or formulafor the derivative. OK, so for the derivative if we want to know the slope of thistangent line at any point call it x we can start with what we do know the goodold slope of the straight line between two points. Pick a second point somewhere onthe curve and draw a straight line (limit definition of derivative) through. And we know how to find the slopeof a straight line with the old-fashioned slope formula, which is a throwback tomiddle school probably. So finding the slope between two points has come back tohaunt you. But you have to know your points. So let’s label them. If thissecond point is some distance away horizontally, let’s call that distance h,then this point has an x coordinate that is h more than x or x + h. We want theactual points the x, y pairs.

 

So for this point, for the x, y pair, the x coordinatewill be x. And the y coordinate we’ll call f(x) which just stands for the yvalue that the function will give us. So that’s that point. And for this otherpoint, for the x, y pair, the x coordinate will be not x but x + h. And the ycoordinate will be not f(x), but f(x) + h. And just humor me. This is all goingsomewhere. We have two points, which is great, because we can use those two pointsin the slope formula that you know and love. OK, so here’s the slope (limit definition of derivative) formulait’s the rise over run or vertical change over the horizontal change. It’s y2 – y1all over the x2 – x1. So let’s use that for our two points. OK, so this is the slope wefound for the secant line. I did some simplifying and you could cancel ‘x’s inthe bottom but this is the slope and it ended up being the difference in the ‘y’sover the horizontal difference, the h.

So it’s the slope of the secant line. Andit’s also known as the difference quotient if you hear that because it’s a quotientof differences. Very original name. This is the slope of the secant line. And weare halfway to the derivative so bear with me. So we are almost to the derivative.And we would be there. We would be done if we had a straight-line graph. A linearfunction, then this slope we found would be right everywhere. It would be enough.But we don’t have that we have a curved graph. And the straight line secant slopethat we found it’s actually an okay approximation for the slope at that pointx.

It’s a rough estimate. And that’s a big part of calculus is estimating something nonlinearwith something linear. So it’s a rough estimate, the slope we found. And it’s notgreat but it’s decent. But let’s not settle for decent. We don’t want anapproximation of the slope here. We want the exact slope, the real slope there andnot some wonky approximation between it and some point nearby. No, we can make itexact if we close in on x by narrowing h to zero and picking a right point that iscloser and closer to the left so that the horizontal distance gets smaller andsmaller. And the closer those two points are together the more (limit definition of derivative) accurate ourestimate is for the slope at that point. And we can make it perfect if we make h sosmall – make h infinitely smaller by taking the limit as h approaches zero. So we’regoing to do a better and better approximation until it’s spot on. So thesecant line becomes the tangent line as this point shifts toward x. By the timeit’s the same point as x, this line is accurate. It’s the tangent line slope. Sothe secant line becomes the tangent line.

The slope of the secant line becomes theslope of the tangent line. In other words, the limit of the secant slope is thetangent line slope when it’s only touching at one point. And so this limit is theslope of the tangent line. And surprise, surprise that limit also defines thederivative. This is the definition of the derivative. So we have finally arrived atthe definition of the derivative. So this is the definition of the derivative. Whenthat limit exists, this defines that that’s the derivative. And this notationyou just read as f'(x). It just means the derivative of f. Also I should say becausethat slope, tangent line slope, can be always changing in different places -it’s variable. It can be variable and that’s why the derivative is (limit definition of derivative)  afunction. It is itself a function just like the original function also a function.Just to reiterate the derivative tells you the slope everywhere – how steep, riseover run, slope, slope number, at the risk of sounding like a broken recordderivative is basically slope.

And to be honest this might not really click andmake sense until you see it used in something like physics where if you’relooking at something moving in time, the position changing in time, a curved graphwould mean that the slope is always changing. The velocity is changing and notconstant because there’s an acceleration. Whereas if you had a straight-line graph,the slope would be the same everywhere because the velocity is not changingbecause there’s no acceleration. But anyway, this is the definition of the derivative. Now the question is.. how do you use it? OK, so say you have tofind the derivative using the definition of the derivative or by the limit process,this is what you use. You use the definition of the derivative but for yourf(x) – whatever you’re given. So to find the derivative f'(x) it’s going to beequal to the limit as h approaches zero. That’s there out front.

Limit as happroaches zero. And then we fill in this part of the formula. f(x) + h means takeyour f, and in place of x, you use x + h, like all of x + h in place of x everywherex appears. So let’s do that. The f(x + h) part… which looks like this3 times (x + h)^2, instead of x^2, plus 12. (limit definition of derivative) Then you subtract f(x), but you’resubtracting all of f(x) just as is however the f(x) looks, but make sureyou use parenthesis when you do the subtraction so you get the right signs. Sowe have minus all of f(x). And then it’s all over h in the formula. Now we can justdo some algebra and simplify. OK,

so here’s all the work to find thederivative. Our answer was 6x but in the work, I foiled. I distributed. I factored.I canceled terms. At one point I factored out an h so that it would cancel with thebottom h. And in the end I took the limit. And I was able to take the limit byplugging in zero for h. And I got 6x which is a beautiful, simple result after allthat work in algebra. The derivative is just 6x. f'(x) is 6x. What that means isthat the derivative of this function 3x^2 + 12 is just 6x. And anywhere inthat function f, the slope would be the number you get by using 6x at any instant,any x. The slope is what you get from 6x at that instant.

So that’s how you findthe derivative using the definition with the limit. Just remember that… that’s theonly tricky part is that this part f(x) + h means that instead of x, you use x + h. Soall of x + h in place of x. And at this point, I should say, just FYI, as a publicservice announcement, (limit definition of derivative) this way with the definition and the limit is good and all.It’s correct and illustrative, but in reality, it’s pretty tedious. It’s a lotof extra algebra and in practice, that’s not really how most people takederivatives. So if you’re taking a lot of derivatives,

all this extra algebra is,what’s the word… a disaster. And there’s a faster, simpler way. And if nothingexplicitly says you have to find it using the definition or by the limit process,you can use the derivative rules, which are a faster, simpler way. Much less extraalgebra. And I have a video on how to find the derivative with the derivative rules.So you can jump to that for that explanation.