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So let's just look at a quick little example here.
So I wrote down a function g(w)=5-(w-10)2.
So let's take the derivative of this function.
So we have to go back and remember our calculus but
I promise I'll at least try to keep it simple.
So the derivative of 5 with respect to w is 0.
And then the derivative of this term here, well if you remember, if you have some
function to some power, well first we drop that power, that becomes a multiplier.
We write that function again, and
the power now is one less than what the power was before.
So two minus one is one.
So I'll just write that explicitly to the power of one,
and then I have to multiply by the derivative of the inside.
The derivative of this little function itself.
So what's the derivative of w-10 with respect to w.
It's just 1.
Okay.
So, writing this a little bit more compactly we have -2w
+ 20 as the derivative.
And so if we're interested in finding the maximum of this equation,
and let me just say how do we know if we're at a maximum or minimum?
In both cases we had derivative equals zero as the solution, well to know if it's
a min or max, either we can know if we're looking at a concave or convex function,
but typically we won't necessarily know that.
We can look at the second derivative of the function.
Okay, but that's just for your information.
You're not gonna need to know that for this course.
Okay, so this function, if
we draw what this function looks like, then maybe we can see this.
So we know that at w equals 10, what's the value of the function?
It's five.
So add w=10, the value of the function is 5 and
if you go and plot this you'll see that the value everywhere else is decreasing
and so it, it has this concave form.
Okay.
So, how do I find this maximum?
Well I take the derivative and I set it equal to zero and I solve for w.
So let's do that.
Let's set derivative = 0.
So I have -2w + 20 = 0.
And I solve for w and I get 10.
Okay, so that actually matches his picture.
Indeed, w is 10 at the maximum of this function.
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