Name: 02-2 衍生物的形式和替代形式（Formal and alternate form of the derivative | Differential Calculus | Khan Academy）
Uploaded: 2022-07-12T12:05:38.000Z
Duration: 4 min 53 s
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I have drawn in red, at the point x equals a.

And we've already seen this with the definition

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.

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

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

We can find the slope of the secant line that

So that would be your change in your vertical,

of x, over the change in the horizontal, which

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, 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

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.

you would just substitute a into your function definition.

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

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

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

as the alternate form of the derivative--

Let's just take another arbitrary point someplace.

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

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

Well, it would be change in the vertical, which

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

Actually, let me do that in that purple color.

Now, how could we get a better and better approximation

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

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.

looking to find the derivative exactly at a.