00:01
Consider the given graph of f of x and g of x below.
00:04
We suppose h of x equals f of g of x.
00:07
Now by chain rule, our h prime of x, this is equal to the outside derivative, that's f prime of g of x, times the inside derivative that's g prime of x.
00:21
So now when x is, let's say, 1, we have h prime of 1 equal to f prime of g of 1.
00:31
This times g prime of 1.
00:34
So looking at our graph, g of 1, that's the value of g when x is 1, and that's going to be approximately 2 .25.
00:44
So we have f prime of 2 .25 times g prime of 1.
00:50
Now, f prime and g prime, these are values of the slopes of the tangent lines to the graphs of f of x and g of x.
01:00
But because f of x and g of x are both lines, then the slopes of the tangent lines will be the same as the slope of the curves f of x and g of x.
01:13
So for f prime of 2 .25, we would need to get the slope of the line at x equals 2 .25, which will be over this side.
01:28
So we will use this line to get the slope of the tangent line.
01:38
And since the slope of the tangent line is a rise overrun, that's going to be a rise of two units.
01:46
If we started at this point, and if we end at this point, since it's a clearer version, than what we have here, so then it would be going to the left two units.
02:02
And going to the left is a negative value, so we would do divided by negative 2.
02:07
So that's for f prime of 2 .25.
02:11
You multiply this by the derivative of g at x equals 1, or the slope of the tangent line to g at x equals 1.
02:22
So the better thing to do is to choose points that are clear enough to get values.
02:28
So we would choose this point again.
02:31
And then this point...