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In this video, I'll show you a slightly more complex example

Â where the slope of the function can be different to different points in the function.

Â Let's start with an example.

Â You have plotted the function f(a) = aÂ².

Â Let's take a look at the point a=2.

Â So aÂ² or f(a) = 4.

Â Less nudge a slightly to the right, so now a=2.001.

Â f(a) which is aÂ² is going to be approximately 4.004.

Â It turns out that the exact value,

Â you call the calculator and figured this out is actually 4.004001.

Â I'm just going to say 4.004 is close enough.

Â So what this means is that when a=2,

Â let's draw this on the plot.

Â So what we're saying is that if a=2,

Â then f(a) = 4 and here is the x and y axis are not drawn to scale.

Â Technically, does vertical height should be much higher than

Â this horizontal height so the x and y axis are not on the same scale.

Â But if I now nudge a to 2.001 then f(a) becomes roughly 4.004.

Â So if we draw this little triangle again,

Â what this means is that if I nudge it to the right by 0.001,

Â f(a) goes up four times as much by 0.004.

Â So in the language of calculus,

Â we say that a slope that is the derivative of f(a) at

Â a=2 is 4 or to write this out of our calculus notation,

Â we say that d/da of f(a) = 4 when a=2.

Â Now one thing about this function f(a) = aÂ²

Â is that the slope is different for different values of a.

Â This is different than the example we saw on the previous slide.

Â So let's look at a different point.

Â If a=5, so instead of a=2,

Â and now a=5 then aÂ²=25, so that's f(a).

Â If I nudge 8 to the right again,

Â it's tiny little nudge to 8,

Â so now a=5.001 then f(a) will be approximately 25.010.

Â So what we see is that by nudging a up by .001,

Â f(a) goes up ten times as much.

Â So we have that d/da f(a) = 10 when

Â a=5 because f(a) goes up ten times as

Â much as a does when I make a tiny little nudge to 8.

Â So one way to see why did derivatives is different at different points is that if you

Â draw that little triangle right at different locations on this,

Â you'll see that the ratio of the height of the triangle over

Â the width of the triangle is very different at different points on the curve.

Â So here, the slope=4 when a=2, a=10, when a=5.

Â Now if you pull up a calculus textbook,

Â a calculus textbook will tell you that d/da of f(a),

Â so f(a) = aÂ²,

Â so that's d/da of aÂ².

Â One of the formulas you find are the calculus textbook is that this thing,

Â the slope of the function aÂ²=2a.

Â Now going to prove this by the way you find this out is that

Â you open up a calculus textbook to

Â the table formulas and they'll tell you that derivative of 2 of aÂ²=2a.

Â And indeed, this is consistent with what we've worked out.

Â Namely, when a=2, the slope of function to a is 2x2=4.

Â And when a=5 then the slope of the function 2xa is 2x5=10.

Â So, if you ever pull up a calculus textbook and you see this formula,

Â that the derivative of aÂ²=2a,

Â all that means is that for any given value of a,

Â if you nudge upward by 0.001 already your tiny little value,

Â you will expect f(a) to go up by 2a.

Â That is the slope or the derivative times

Â other much you had nudged to the right the value of a.

Â Now one tiny little detail,

Â I use these approximate symbols here and this wasn't exactly 4.004,

Â there's an extra .001 in out there.

Â It turns out that this extra .001,

Â this little thing here is because we were nudging a to the right by 0.001,

Â if we're instead nudging it to the right by

Â this infinitesimally small value then this extra every term will go

Â away and you find that the amount that f(a) goes out is exactly equal

Â to the derivative times the amount that you nudge a to the right.

Â And the reason why is not 4.004 exactly is because derivatives are defined using

Â this infinitesimally small nudges to a rather than 0.001 which is not.

Â And while 0.001 is small,

Â it's not infinitesimally small.

Â So that's why the amount that f(a) went up isn't exactly given

Â by the formula but it's only a kind of approximately given by the derivative.

Â To wrap up this video,

Â let's just go through a few more quick examples.

Â The example you've already seen is that if f(a) = aÂ² then

Â the calculus textbooks formula table will tell you that the derivative is equal to 2a.

Â And so the example we went through was it if (a) = 2,

Â f(a) = 4, and we nudge a,

Â since it's a little bit bigger than f(a) is about

Â 4.004 and so f(a) went up four times as much and indeed when a=2,

Â the derivatives is equal to 4.

Â Let's look at some other examples.

Â Let's say, instead the f(a) = aÂ³.

Â If you go take half of this textbook and look up the table formulas,

Â you see that the slope of this function, again,

Â the derivative of this function is equal to 3aÂ².

Â So you can get this formula out of the calculus textbook.

Â So what this means?

Â So the way to interpret this is as follows.

Â Let's take a=2 as an example again.

Â So f(a) or aÂ³=8, that's two-part three.

Â So we give a a tiny little nudge,

Â you find that f(a) is about 8.012 and feel free to check this.

Â Take 2.001 tp part three,

Â you find this is very close to 8.012.

Â And indeed, when a=2 that's 3x2Â² does equal to 3x4,

Â you see that's 12.

Â So the derivative formula predicts that if you nudge a to the right by tiny little bit,

Â f(a) should go up 12 times as much.

Â And indeed, this is true when a went up by .001,

Â f(a) went up 12 times as much by .012.

Â Just one last example and then we'll wrap up.

Â Let's say that f(a) is equal to the log function.

Â So on the right log of a,

Â I'm going to use this as the base e logarithm.

Â So some people write that as log(a).

Â So if you go to calculus textbook,

Â you find that when you take the derivative of log(a).

Â So this is a function that just looks like that,

Â the slope of this function is given by 1/a.

Â So the way to interpret this is that if a has any value then let's just keep

Â using a=2 as an example and you nudge a to the right by .001,

Â you would expect f(a) to go up by

Â 1/a that is by the derivative times the amount that you increase a.

Â So in fact, if you pull up a calculator,

Â you find that if a=2,

Â f(a) is about 0.69315 and if you

Â increase f and if you increase a to 2.001 then f(a) is about 0.69365,

Â this has gone up by 0.0005.

Â And indeed, if you look at the formula for the derivative when a=2,

Â d/da f(a) = 1/2.

Â So this derivative formula predicts that if you pump up a by .001,

Â you would expect f(a) to go up by only 1/2 as much and 1/2 of .001

Â is 0.0005 which is exactly what we got.

Â Then when a goes up by .001 going from a=2/3,

Â a=2.01, f(a) goes up by half as much.

Â So, the answers are going up by approximately .0005.

Â So if we draw that little triangle if you will is that if on

Â the horizontal axis just goes up by .001 on the vertical axis,

Â log(a) goes up by half of that so .0005.

Â And so that 1/a or 1/2 in this case,

Â 1a=2 that's just the slope of this line when a=2.

Â So that's it for derivatives.

Â There just two take home messages from this video.

Â First is that the derivative of the function just means the slope of

Â a function and the slope of a function

Â can be different at different points on the function.

Â In our first example where f(a) = 3a those a straight line.

Â The derivative was the same everywhere,

Â it was three everywhere.

Â For other functions like f(a) = aÂ² or f(a) = log(a),

Â the slope of the line varies.

Â So, the slope or the derivative can be different at different points on the curve.

Â So that's a first take away.

Â Derivative just means slope of a line.

Â Second takeaway is that if you want to look up the derivative of a function,

Â you can flip open your calculus textbook or look up Wikipedia and

Â often get a formula for the slope of these functions at different points.

Â So that, I hope you have an intuitive understanding of derivatives or slopes of lines.

Â Let's go into the next video.

Â We'll start to talk about the computation drop and how to

Â use that to compute derivatives of more complex.

Â