### Video Transcript

Find the first derivative of the function ๐ฆ equals five ๐ฅ squared plus two multiplied by nine ๐ฅ cubed plus six ๐ฅ plus four.

So with this type of question, the first thing we want to do is look to expand the parentheses. And thatโs before we do any differentiating. So first of all, we have five ๐ฅ squared multiplied by nine ๐ฅ cubed. Thatโs gonna give us 45๐ฅ power of five. Thatโs because if we multiply five and nine, we get 45. And then, we apply one of our exponent laws. And that is that if we have ๐ฅ power of ๐ multiplied by ๐ฅ power of ๐, so weโve got the same bases here, then what we do is we add the exponents.

So in that case, we had ๐ฅ squared multiplied by ๐ฅ cubed. So we add two and three, which gives us ๐ฅ power of five. So we get 45๐ฅ to the power of five. Then, we have five ๐ฅ squared multiplied by six ๐ฅ, which gives us 30๐ฅ cubed. Then, five ๐ฅ squared multiplied by four, which gives us 20๐ฅ squared.

So now, what weโre gonna do is multiply the second parenthesis by the positive two in the first parenthesis. So we get plus 18๐ฅ cubed. And thatโs because we had two multiplied by nine ๐ฅ cubed. Then, plus 12๐ฅ because we have two multiplied by six ๐ฅ. Then finally, plus eight. And thatโs because we have two multiplied by four.

So now, we have the function in this form. What we want to do is collect any like terms. But the only like terms we have are our ๐ฅ cubed terms. So we got 30๐ฅ cubed and 18๐ฅ cubed, which gives us 48๐ฅ cubed. So we got 45๐ฅ power of five plus 48๐ฅ cubed plus 20๐ฅ squared plus 12๐ฅ plus eight.

So now, what weโre gonna do is weโre gonna differentiate this to find our first derivative. So our first term is going to be 225๐ฅ to the power of four. And just remind us how we did that. So how we differentiate. What we do is we multiply the exponent by the coefficient. So we have five multiplied by 45, which gives us our 225. And then, we reduce the exponent by one, so five minus one, which gives us four. So we get 225๐ฅ to the power of four.

Then, using the same method, we get the second term, which is 144๐ฅ squared. Thatโs because three multiplied by 48 is 144. Then, we reduce the power or the exponent from three to two. And then, the next term is 40๐ฅ. Thatโs cause we reduced the exponent by one to give us ๐ฅ, cause ๐ฅ to the power of one. And we multiplied two by 20, which gives us our 40. And then, the final term is just 12. And thatโs because if we differentiate 12๐ฅ, we just get 12, because if we reduce the exponent on ๐ฅ from one to zero, it gives us ๐ฅ power of zero, which should just be one. And if we differentiate eight, we just get zero.

So therefore, we can say that the first derivative of the function ๐ฆ equals five ๐ฅ squared plus two multiplied by nine ๐ฅ cubed plus six ๐ฅ plus four is 225๐ฅ to the power of four plus 144๐ฅ squared plus 40๐ฅ plus 12.