00:03
All right, you fans of ugly algebra, have i got one for you? we want to differentiate this function, and we actually have to apply the chain roll twice, because this whole thing is a composite of two functions, right? we have this outside exponent here.
00:24
So our function is something squared.
00:29
That's the outside function.
00:31
And the derivative of that is, of course, two times whatever.
00:36
That inside is and then subtract one from the power.
00:39
Very easy to do.
00:41
However, you'll notice that our inside function, so when we do the chain rule initially, this is our inside function.
00:54
Within the inside function is another function that's a composite that's going to require the chain rule.
01:01
So when we get to the part of the question or the part of the calculation where we have to compute the derivative, of that inside function, we're going to have to apply the chain rule again.
01:12
So just be careful.
01:14
You know, i like to separate things as much as i can, and it's very easy to overlook some things.
01:21
All right, so let's see what we get here.
01:25
F prime of x then is going to be negative one times.
01:31
Sorry, let's do the wrong thing.
01:33
Outside function is going to be two times the quantity 6x, plus x to the fifth to the negative first power plus x, then we subtract one from the exponent, and that's a one.
01:50
I will write it there, but i'm going to take that off once we simplify a little bit.
01:55
And then we do times the derivative of the inside function.
02:01
And the inside function is, again, 6x plus x to the fifth to the negative first plus x.
02:12
And that entails using the chain rule again.
02:17
So we want to take the derivative of this whole thing.
02:20
It's term by term.
02:22
So obviously the derivative of x is just one.
02:25
So we're going to have a plus one at the end of this thing.
02:29
It's this part that's going to require the chain rule again.
02:32
So chain rule tells us it's going to be the derivative of the outside function.
02:39
So it's going to be negative 1 times 6x plus x to the fifth to the negative.
02:45
Two times the derivative of the inside function...