Video: Differentiating Polynomial Functions in the Factored Form

Find the first derivative of the function 𝑦 = (π‘₯Β² + 8)(3π‘₯Β³ βˆ’ 8π‘₯ + 6).

03:52

Video Transcript

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

Finding the first derivative is another way of saying differentiate or find the function 𝑑𝑦 by 𝑑π‘₯. If 𝑦 is equal to π‘Ž multiplied by π‘₯ to the power of 𝑛, then 𝑑𝑦 by 𝑑π‘₯, the differential, is equal to 𝑛 multiplied by π‘Ž multiplied by π‘₯ to the power of 𝑛 minus one.

In our question, we’re told that 𝑦 is equal to π‘₯ squared plus eight multiplied by three π‘₯ cubed minus eight π‘₯ 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 π‘₯ squared and then multiply everything inside the second bracket by eight. π‘₯ squared multiplied by three π‘₯ cubed is equal to three π‘₯ to the power of five. Remember to add the exponents or powers. Two plus three is equal to five. π‘₯ squared multiplied by negative eight π‘₯ is equal to negative eight π‘₯ cubed. And finally, π‘₯ squared multiplied by six is equal to six π‘₯ squared.

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

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

We are now going to differentiate each of the terms individually to work out the first derivative. The differential of three π‘₯ to the power of five is 15π‘₯ 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 16π‘₯ cubed gives us 48π‘₯ squared. The differential of six π‘₯ squared is equal to 12π‘₯, as two multiplied by six is equal to 12. Differentiating negative 64π‘₯ gives us negative 64, as one multiplied by 64 is equal to 64. Subtracting one from the power gives us π‘₯ 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 π‘₯ squared plus eight multiplied by three π‘₯ cubed minus eight π‘₯ plus six is 15π‘₯ to the power of four plus 48π‘₯ squared plus 12π‘₯ minus 64.

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