00:03
We need to find a function f of x and a number for a such that f prime the derivative of f at a is equal to this limit.
00:17
Now for a function f of x and a value a, so x equals some number a, the derivative of our function of x when x is a, is equal to the limit of f of a plus h minus f of a, all divided by h as we take the limit as h approaches zero.
01:13
So if we have some function f of x and some number a, f prime of a equals the limit of f of a plus h minus f of a over h as h goes towards zero.
01:25
We need to find, excuse me, we need to find a function f of x and the number a so that when we find f prime at a using the definition of the derivative, we will get this limit for our function f of x and for our number of a.
01:51
Well, let's start comparing this to this.
01:59
Plus h function of a plus h.
02:03
Here we have e to the negative 2 plus h.
02:08
A plus h, a plus h, negative 2 plus h.
02:11
That's starting to make me think that maybe a is negative 2.
02:18
Now, here the function might be a little bit easier to see because you see this e here and maybe you think of e to the x.
02:31
So let's write down possibly a is the number negative 2.
02:43
Possibly our function f of x is e to the x.
02:49
Now, if it turns out that f of x really is e to the x and a equals negative 2, then we're done because this is what we're actually supposed to be finding.
02:59
A function f of x and a number a, such that f prime at a equals this limit.
03:06
Well, using f of x equals e to the x and using a equal to negative 2, let's use the definition for the derivative, f prime of a, equals this, to see if these are indeed to correct f of x and correct a.
03:27
So we're going to apply this definition of the derivative for this function and this value of a.
03:36
So if f of x equals e to the x and a is negative 2, f prime at a, f prime at negative 2, would equal the limit as h approaches 0 of f of a plus h.
04:06
Now, we're letting a be negative 2.
04:10
We're letting f of x be e to the x...