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• So let's think about how we could

• find the slope of the tangent line

• to this curve right over here, so what

• I have drawn in red, at the point x equals a.

• And we've already seen this with the definition

• of the derivative.

• We could try to find a general function that gives us

• the slope of the tangent line at any point.

• So let's say we have some arbitrary point.

• Let me define some arbitrary point x right over here.

• Then this would be the point x comma f of x.

• And then we could take some x plus h.

• So let's say that this right over here is the point

• x plus h.

• And so this point would be x plus h, f of x plus h.

• We can find the slope of the secant line that

• goes between these two points.

• which would be f of x plus h minus f

• of x, over the change in the horizontal, which

• would be x plus h minus x.

• And these two x's cancel.

• So this would be the slope of this secant line.

• And then if we want to find the slope of the tangent line at x,

• we would just take the limit of this expression

• as h approaches 0.

• As h approaches 0, this point moves towards x.

• And that slope of the secant line between these two

• is going to approximate the slope of the tangent line at x.

• And so this right over here, this we would say

• is equal to f prime of x.

• This is still a function of x.

• You give me an arbitrary x where the derivative is defined.

• I'm going to plug it into this, whatever this ends up being.

• It might be some nice, clean algebraic expression.

• Then I'm going to give you a number.

• So for example, if you wanted to find--

• you could calculate this somehow.

• Or you could even leave it in this form.

• And then if you wanted f prime of a,

• you would just substitute a into your function definition.

• And you would say, well, that's going

• to be the limit as h approaches 0 of-- every place you see

• an x, replace it with an a. f of-- I'll

• stay in this color for now-- blank plus h minus f of blank,

• all of that over h.

• And I left those blanks so I could write the a in red.

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

• So this is the derivative evaluated at a.

• So this is one way to find the slope of the tangent line

• when x equals a.

• Another way-- and this is often used

• as the alternate form of the derivative--

• would be to do it directly.

• So this is the point a comma f of a.

• Let's just take another arbitrary point someplace.

• So let's say this is the value x.

• This point right over here on the function would be x comma

• f of x.

• And so what's the slope of the secant line between these two

• points?

• Well, it would be change in the vertical, which

• would be f of x minus f of a, over change in the horizontal,

• over x minus a.

• Actually, let me do that in that purple color.

• Over x minus a.

• Now, how could we get a better and better approximation

• for the slope of the tangent line here?

• Well, we could take the limit as x approaches a.

• As x gets closer and closer and closer to a,

• the secant line slope is going to better and better

• and better approximate the slope of the tangent line,

• this tangent line that I have in red here.

• So we would want to take the limit as x approaches a here.

• Either way, we're doing the exact same thing.

• We have an expression for the slope of a secant line.

• And then we're bringing those x values of those points

• closer and closer together.

• So the slopes of those secant lines better and better

• and better approximate that slope of the tangent line.

• And at the limit, it does become the slope of the tangent line.

• That is the definition of the derivative.

• So this is the more standard definition of a derivative.

• It would give you your derivative as a function of x.

• And then you can then input your particular value of x.

• Or you could use the alternate form of the derivative.

• If you know that, hey, look, I'm just

• looking to find the derivative exactly at a.

• I don't need a general function of f.

• Then you could do this.

• But they're doing the same thing.

So let's think about how we could

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# Formal and alternate form of the derivative | Differential Calculus | Khan Academy

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yukang920108 posted on 2022/07/12
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