Video: Differentiating Polynomial Functions in the Factored Form

Video Transcript

Find the first derivative of the function 𝑦 is equal to 𝑥 squared plus eight multiplied by three 𝑥 cubed minus eight 𝑥 plus six.

Finding the first derivative is another way of saying differentiate or find the function 𝑑𝑦 by 𝑑𝑥. If 𝑦 is equal to 𝑎 multiplied by 𝑥 to the power of 𝑛, then 𝑑𝑦 by 𝑑𝑥, the differential, is equal to 𝑛 multiplied by 𝑎 multiplied by 𝑥 to the power of 𝑛 minus one.

In our question, we’re told that 𝑦 is equal to 𝑥 squared plus eight multiplied by three 𝑥 cubed minus eight 𝑥 plus six. The easiest way to deal with this problem is to firstly expand the parentheses. We’ll multiply everything inside the second parenthesis by 𝑥 squared and then multiply everything inside the second bracket by eight. 𝑥 squared multiplied by three 𝑥 cubed is equal to three 𝑥 to the power of five. Remember to add the exponents or powers. Two plus three is equal to five. 𝑥 squared multiplied by negative eight 𝑥 is equal to negative eight 𝑥 cubed. And finally, 𝑥 squared multiplied by six is equal to six 𝑥 squared.

We then need to multiply all three terms by eight. Eight multiplied by three 𝑥 cubed is equal to 24𝑥 cubed. Eight multiplied by negative eight 𝑥 is negative 64𝑥. And finally, eight multiplied by six is equal to 48.

We then need to collect the like terms. Negative eight 𝑥 cubed plus 24𝑥 cubed is equal to 16𝑥 cubed. This means that 𝑦 is equal to three 𝑥 to the power of five plus 16𝑥 cubed plus six 𝑥 squared minus 64𝑥 plus 48.

We are now going to differentiate each of the terms individually to work out the first derivative. The differential of three 𝑥 to the power of five is 15𝑥 to the power of four, as five multiplied by three is equal to 15 and subtracting one from the power gives us four. In the same way, differentiating 16𝑥 cubed gives us 48𝑥 squared. The differential of six 𝑥 squared is equal to 12𝑥, as two multiplied by six is equal to 12. Differentiating negative 64𝑥 gives us negative 64, as one multiplied by 64 is equal to 64. Subtracting one from the power gives us 𝑥 to the power of zero, and we know that any number to the power of zero is equal to one. Finally, differentiating 48 gives us zero, as the differential of any number is equal to zero.

Therefore, the first derivative of the function 𝑦 is equal to 𝑥 squared plus eight multiplied by three 𝑥 cubed minus eight 𝑥 plus six is 15𝑥 to the power of four plus 48𝑥 squared plus 12𝑥 minus 64.