Question Video: Finding the First Derivative of a Polynomial Function | Nagwa Question Video: Finding the First Derivative of a Polynomial Function | Nagwa

# Question Video: Finding the First Derivative of a Polynomial Function Mathematics • Second Year of Secondary School

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Find the first derivative of the function ๐ฆ = (5๐ฅยฒ + 2)(9๐ฅยณ + 6๐ฅ + 4).

03:10

### Video Transcript

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

So with this type of question, the first thing we want to do is look to expand the parentheses. And thatโs before we do any differentiating. So first of all, we have five ๐ฅ squared multiplied by nine ๐ฅ cubed. Thatโs gonna give us 45๐ฅ power of five. Thatโs 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 ๐ฅ power of ๐ multiplied by ๐ฅ power of ๐, so weโve got the same bases here, then what we do is we add the exponents.

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

So now, what weโre gonna do is multiply the second parenthesis by the positive two in the first parenthesis. So we get plus 18๐ฅ cubed. And thatโs because we had two multiplied by nine ๐ฅ cubed. Then, plus 12๐ฅ because we have two multiplied by six ๐ฅ. Then finally, plus eight. And thatโs 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 ๐ฅ cubed terms. So we got 30๐ฅ cubed and 18๐ฅ cubed, which gives us 48๐ฅ cubed. So we got 45๐ฅ power of five plus 48๐ฅ cubed plus 20๐ฅ squared plus 12๐ฅ plus eight.

So now, what weโre gonna do is weโre gonna differentiate this to find our first derivative. So our first term is going to be 225๐ฅ 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 225๐ฅ to the power of four.

Then, using the same method, we get the second term, which is 144๐ฅ squared. Thatโs 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 40๐ฅ. Thatโs cause we reduced the exponent by one to give us ๐ฅ, cause ๐ฅ 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 thatโs because if we differentiate 12๐ฅ, we just get 12, because if we reduce the exponent on ๐ฅ from one to zero, it gives us ๐ฅ 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 ๐ฆ equals five ๐ฅ squared plus two multiplied by nine ๐ฅ cubed plus six ๐ฅ plus four is 225๐ฅ to the power of four plus 144๐ฅ squared plus 40๐ฅ plus 12.

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