### Video Transcript

Given that π¦ equals π₯ minus seven multiplied by four π₯ plus seven and π§ equals π₯ squared plus five π₯ plus nine, determine π squared π¦ over ππ₯ squared plus π squared π§ over ππ₯ squared.

Now, in order to actually solve this problem, what weβre gonna need to do is actually differentiate each of our functions and then differentiate them again to find the second-order derivative. So Iβm gonna start with our first function, which is π¦ is equal to π₯ minus seven multiplied by four π₯ plus seven.

Well, the first thing weβre actually gonna have to do here is expand the parentheses. So the first term weβre gonna get is π¦ is equal to four π₯ squared cause weβve got π₯ multiplied by four π₯, which gives us four π₯ squared. Then, we get plus seven π₯ because we have π₯ multiplied by positive seven minus 28π₯ cause we had negative seven multiplied by four π₯. And then, finally, weβve got minus 49. So weβve got four π₯ squared plus seven π₯ minus 28π₯ minus 49.

So what we can do next is actually simplify this by collecting like terms. So weβre gonna get π¦ is equal to four π₯ squared minus 21π₯ minus 49. Okay, fab, so this is out in a form, which is much easier to differentiate.

So what weβre gonna do first of all is actually find the first-order derivative. And when we actually differentiate four π₯ squared minus 21π₯ minus 49, we get eight π₯ minus 21. And just a quick recap of why we got that, weβre gonna look at the first term. Well, if weβve got four multiplied by two and thatβs because you have the coefficient multiplied by the exponent and then π₯ to the power of and then we got two minus one and thatβs cause you subtract one from the exponent, so that leaves us with eight π₯.

So therefore, thatβs how we actually differentiated each term. So weβve got eight π₯ minus 21 as our first-order derivative. So therefore, what we want to do now is actually find the second-order derivative.

And the way weβre gonna do that is by finding π squared π¦ over ππ₯ squared. And to do that, we actually differentiate eight π₯ minus 21. And when we do that, weβre just gonna get the result as eight. And thatβs because eight π₯ differentiates to just eight because again if you multiply the exponent by the coefficient, youβd have eight by one, which gives us eight. And then, you reduce the exponent of π₯ by one. So you go from π₯ to the power of one to π₯ to the power of zero, which would just give us one. So weβre just left with eight because negative 21 differentiates to zero.

So great! Weβve actually found π squared π¦ over ππ₯ squared for our first function. Now, letβs move on to our second function.

So looking at our second function, weβve got π§ is equal to π₯ squared plus five π₯ plus nine. So therefore, if we actually differentiate this, what weβre gonna get is ππ§ ππ₯ is equal to two π₯ plus five. And thatβs because again we use the same method that we had for the previous part. So if we differentiate the first term, weβd have the coefficient which is one multiplied by two, the exponent gives us two. And then, you reduce the exponent by one. So we just get two π₯.

Okay, great, so we found that. So again, what we need to do is actually find the second-order derivative. So in order to do that, so weβre gonna find π squared π§ over ππ₯ squared. So we just differentiate two π₯ plus five. And when we do that, weβre just left with two.

So great, weβve now actually found the second derivative of π¦ equals π₯ minus seven multiplied by four π₯ plus seven and the second derivative of π§ equals π₯ squared plus five π₯ plus nine.

So now, letβs look back at the question what we need to do next. Well, we can see that actually we need to add our results together.

So therefore, what we can say is that given that π¦ equals π₯ minus seven multiplied by four π₯ plus seven and π§ equals π₯ squared plus five π₯ plus nine, then π squared π¦ over ππ₯ squared plus π squared π§ over ππ₯ squared is gonna be equal to two plus eight, which is equal to 10.