### Video Transcript

Determine the first derivative of the function ๐ฆ equals two ๐ฅ multiplied by nine ๐ฅ squared minus three ๐ฅ plus 10๐ฅ.

So now, the first stage we actually need to complete is to expand our parentheses. So in order to do this, what weโre gonna do is multiply out two ๐ฅ by each term. So weโre gonna start with two ๐ฅ multiplied by nine ๐ฅ squared which will give us 18๐ฅ cubed. And then, weโre gonna have two ๐ฅ multiplied by negative three ๐ฅ. Very important here to make sure that we do include the sign. So negative, so itโs two ๐ฅ multiplied by negative three ๐ฅ. So therefore, our next term is gonna be minus six ๐ฅ squared. And finally, we add on our 10๐ฅ.

Okay, great. So we now have the function ๐ฆ equals 18๐ฅ cubed minus six ๐ฅ squared plus 10๐ฅ. So now that weโve actually expanded the parentheses, what we can do is actually differentiate each of our terms individually. So before we do this, letโs remind ourselves how we actually differentiate.

So if we look at differentiation and weโve got a function in the form ๐๐ฅ to the power of ๐, then the derivative of this function is gonna be equal to ๐๐๐ฅ to the power of ๐ minus one. So what weโve done is weโve multiplied the exponent by the coefficient. And then what weโve done is reduce the exponent itself by one. So we get ๐๐๐ฅ to the power of ๐ minus one. Okay, great. So now we kind of recapped this. Letโs use it to actually differentiate each of our terms.

So Iโm gonna go through them each step by step, just so we can see exactly what each of our terms is going to be. So first of all, if we differentiate 18๐ฅ cubed, this is gonna be equal to 18 multiplied by three. So the coefficient multiplied by the exponent. And then ๐ฅ to the power of three minus one which would give us 54๐ฅ squared. Okay, great. We found our first term in the derivative. Itโs also worth noting at this point, Iโve put our first derivative as ๐๐ฆ ๐๐ฅ. But with the notation, you could also see it as ๐ฆ prime or ๐ ๐ฅ prime. They both mean the same thing. It means the first derivative.

Okay, great. So now, letโs move on to our second term. So for our second term, weโre gonna differentiate negative six ๐ฅ squared. Iโve kept the negative in because actually itโs the sign thatโs relevant to this term. However, it wonโt actually affect the differentiation itself. And this is gonna give us negative six multiplied by two because thatโs how again our coefficient multiplied by our exponent. And then ๐ฅ to the power of two minus one because we reduce one from our exponent which will give us negative 12๐ฅ. Okay, great. We found our second term.

So therefore, we can now move on to our last term. Well, our last term is just gonna give us plus 10. And then just to remind ourselves how we actually got that, weโve got 10 multiplied by one because weโve got our coefficient 10. And then the exponent will be one, if itโs ๐ฅ on its own. And then ๐ฅ to the power of one minus one. And if you think about that, that will give us ๐ฅ to the power of zero. And we know anything to the power of zero is just equal to one. So therefore, weโre just left with 10.

So therefore, we can say that the first derivative of the function ๐ฆ equals two ๐ฅ multiplied by nine ๐ฅ squared minus three ๐ฅ plus 10๐ฅ is equal to 54๐ฅ squared minus 12๐ฅ plus 10.