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So let's think about how

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we could find the slope of the tangent line

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to this curve right over here, so what I've drawn in red-

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at the point x=a. We've already seen this with the definition of the derivative.

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We can try to find a general function that gives us the slope of the tangent line at any point

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So let's see we have some arbitrary point - let's define some arbitrary point x right over here

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then this would be the point (x,f(x)) and then we could take some x+h - so let's say that this right over here

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is the point x+h, so this point

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would be (x+h,f(x+h)), we can find the slope of

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the secant line, that goes between these two points,

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so that would be your change in your vertical,

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which would be f(x+h), f of x plus h,

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minus f(x), minus f(x), over the change in the horizontal,

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which would be x+h minus x, minus x;

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and these two x's cancel, so this would be the slope of this secant line,

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and if we wanna find the slope of the tangent line at x - the slope of the tangent line at x-

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we would just take the limit - the limit of this expression -

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as h approaches 0, as h approaches 0, this point moves toward x, and that slope of the

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secant line between these two is going to approximate the slope of the tangent at x

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And so this right over here - this we would say is equal to f'(x) [f-prime of x]

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This is still - this is a function of x! You give me an arbitrary x, where the derivative is defined,

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I'm gonna plug it into this, whatever this ends up being, it might be some clean algebraic expression,

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and then I'm gonna give you a number. So, for example, if you wanted to find

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you, you could calculate this somehow or you could even leave it in this form, and then if you wanted f'(a)

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[f-prime of a], you would just substitute a into your function definition,

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and you would say, well that's going to be the limit as h approaches 0

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of every place you see an x or every place you see an a. I'll stay in this colour for now...

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Blank plus h minus, minus - minus f of blank, minus f of blank - all of that over

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all of that over h. And I left those blank so I could write the a - I could write the a in red.

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Notice every place where I had an x before, it's now an a.

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So this is the derivative evaluated at a. So this is one way to find the slope of the tangent line when x=a.

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Another way - this often viewed as the alternate form of the derivative - would be to do it directly.

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So this is the point (a,f(a)), let's just take another arbitrary point [unintelligible] someplace -

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so let's say this is the value x - this point right over here on the function would be (x,f(x)) -

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So what's the slope of the secant line between these two points? It would be the change in the vertical,

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which would be f(x) minus f(a) - minus f(a)- over a change in the horizontal.

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Over x-a. So let me do that in a purple colour - over x minus a.

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Now, how could we get a better approximation of the slope of the tangent line here?

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Well, we could take the limit as x approaches a, as x gets closer and closer and closer to a,

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the secant line slope is going to better and better and better approximate the slope of the tangent line.

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This tangent line that I have in red here. So we would wanna take the limit

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the limit - as, as x approaches a here. Either way, we're doing a very similar -

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we're doing the exact same thing - we're finding the - we're taking the s- we're, we're having an expression

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for the slope of a secant line, and then we're bringing those x values of those points closer and closer together,

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closer and closer together, so the slopes of those secant lines better and better and better

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approximate that slope of the tangent line.

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And at the limit, it does become the slope of the tangent line, that is the derivative of - that is the definition of

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so this is the cl- the more kind of standard definition of the derivative, if would give you your derivative as a function

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of x, and then you can then input your x if you wan- your, your uh - the particular value of x, or you can

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use the alternate form of the derivative - if you know that hey, look, I'm just looking to find the derivative exactly at a,

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I don't need a general function of f, then you could do this. But they're doing the same thing.