### Video Transcript

If the line ๐ฆ equals three ๐ฅ plus nine is tangent to the graph of the function ๐ at 215, what is ๐ prime two?

So if we look at the question here, we look at the information weโve got, it says that we have a line which is ๐ฆ is equal to three ๐ฅ plus nine and itโs tangent to a graph. Okay, what does this actually mean? Well Iโve drawn us a little sketch to help us.

This little sketch actually just shows any function and then a line that itโs tangent with at a point. So what Iโve actually done with this sketch of โ Iโve just picked a random function and line โ is just demonstrate a relationship we have, that at this point, which Iโve marked on, the slope of the function and the line are going to be equal.

And this is gonna be very important cause weโre gonna use this to actually solve the problem that weโre doing now. So thinking about slope, what weโre gonna first do is have a look at our line which is ๐ฆ is is equal to three ๐ฅ plus nine. But what we can actually see is that our line is in the form ๐ฆ is equal to ๐๐ฅ plus ๐ where ๐ is our slope and ๐ is our ๐ฆ-intercept.

So therefore, if we take a look at the equation of our line, we can see that our slope or our ๐ is gonna be equal to three. And this is because this is our coefficient of ๐ฅ. Okay, so now weโve got the slope at this point of our tangent. Itโs also worth noting at this point that we said that this is the tangent to the graph of the function at the point two, 15.

So we know that actually when weโre looking at the function itself, this is the point that weโre gonna be using. So now weโre gonna move on to the next part of the question. What is ๐ prime two? Well ๐ prime ๐ฅ actually means the slope function. So again, you could seeing it as ๐๐ฆ ๐๐ฅ.

Well thereโs various ways that actually can be represented. But what weโre trying to say that is what is the slope function when ๐ฅ is equal to two; so i.e., what is the slope of our function at the point where ๐ฅ is equal to two? So itโs at this point we can actually go back to the relationship we looked at earlier.

So what the relationship tells us is that at the point where a tangent actually touches our function, the slope of the function and the tangent of a line at that point is gonna be equal. So therefore, if you bring it into our context, we can say that at the point two, 15, the slope of ๐ฆ equals three ๐ฅ plus nine and ๐ ๐ฅ are going to be equal.

So therefore, we can say that ๐ prime two or the value of the slope function when ๐ฅ is equal to two is gonna be equal to the slope of ๐ฆ equals three ๐ฅ plus nine, because the point when the slope function has a value of ๐ฅ equal two is the same as when we have the point where our tangent and function actually meet because itโs two, 15 so the ๐ฅ-value is two.

So therefore, we can say that the value of ๐ prime two must be equal to three because three was actually the slope that we found earlier because we had ๐ is equal to three of our tangent three ๐ฅ plus nine.