Question Video: Finding the First Derivative of a Polynomial Function Mathematics • Higher Education

Find the first derivative of the function 𝑦 = (5𝑥² + 2)(9𝑥³ + 6𝑥 + 4).

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Video Transcript

Find the first derivative of the function 𝑦 equals five 𝑥 squared plus two multiplied by nine 𝑥 cubed plus six 𝑥 plus four.

So with this type of question, the first thing we want to do is look to expand the parentheses. And that’s before we do any differentiating. So first of all, we have five 𝑥 squared multiplied by nine 𝑥 cubed. That’s gonna give us 45𝑥 power of five. That’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 𝑥 power of 𝑎 multiplied by 𝑥 power of 𝑏, so we’ve got the same bases here, then what we do is we add the exponents.

So in that case, we had 𝑥 squared multiplied by 𝑥 cubed. So we add two and three, which gives us 𝑥 power of five. So we get 45𝑥 to the power of five. Then, we have five 𝑥 squared multiplied by six 𝑥, which gives us 30𝑥 cubed. Then, five 𝑥 squared multiplied by four, which gives us 20𝑥 squared.

So now, what we’re gonna do is multiply the second parenthesis by the positive two in the first parenthesis. So we get plus 18𝑥 cubed. And that’s because we had two multiplied by nine 𝑥 cubed. Then, plus 12𝑥 because we have two multiplied by six 𝑥. Then finally, plus eight. And that’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 𝑥 cubed terms. So we got 30𝑥 cubed and 18𝑥 cubed, which gives us 48𝑥 cubed. So we got 45𝑥 power of five plus 48𝑥 cubed plus 20𝑥 squared plus 12𝑥 plus eight.

So now, what we’re gonna do is we’re gonna differentiate this to find our first derivative. So our first term is going to be 225𝑥 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 225𝑥 to the power of four.

Then, using the same method, we get the second term, which is 144𝑥 squared. That’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 40𝑥. That’s cause we reduced the exponent by one to give us 𝑥, cause 𝑥 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 that’s because if we differentiate 12𝑥, we just get 12, because if we reduce the exponent on 𝑥 from one to zero, it gives us 𝑥 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 𝑦 equals five 𝑥 squared plus two multiplied by nine 𝑥 cubed plus six 𝑥 plus four is 225𝑥 to the power of four plus 144𝑥 squared plus 40𝑥 plus 12.

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