WEBVTT
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Find the first derivative of the function π¦ is equal to π₯ squared plus eight multiplied by three π₯ cubed minus eight π₯ plus six.
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Finding the first derivative is another way of saying differentiate or find the function ππ¦ by ππ₯.
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If π¦ is equal to π multiplied by π₯ to the power of π, then ππ¦ by ππ₯, the differential, is equal to π multiplied by π multiplied by π₯ to the power of π minus one.
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In our question, weβre told that π¦ is equal to π₯ squared plus eight multiplied by three π₯ cubed minus eight π₯ plus six.
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The easiest way to deal with this problem is to firstly expand the parentheses.
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Weβll multiply everything inside the second parenthesis by π₯ squared and then multiply everything inside the second bracket by eight.
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π₯ squared multiplied by three π₯ cubed is equal to three π₯ to the power of five.
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Remember to add the exponents or powers.
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Two plus three is equal to five.
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π₯ squared multiplied by negative eight π₯ is equal to negative eight π₯ cubed.
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And finally, π₯ squared multiplied by six is equal to six π₯ squared.
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We then need to multiply all three terms by eight.
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Eight multiplied by three π₯ cubed is equal to 24π₯ cubed.
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Eight multiplied by negative eight π₯ is negative 64π₯.
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And finally, eight multiplied by six is equal to 48.
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We then need to collect the like terms.
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Negative eight π₯ cubed plus 24π₯ cubed is equal to 16π₯ cubed.
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This means that π¦ is equal to three π₯ to the power of five plus 16π₯ cubed plus six π₯ squared minus 64π₯ plus 48.
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We are now going to differentiate each of the terms individually to work out the first derivative.
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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.
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In the same way, differentiating 16π₯ cubed gives us 48π₯ squared.
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The differential of six π₯ squared is equal to 12π₯, as two multiplied by six is equal to 12.
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Differentiating negative 64π₯ gives us negative 64, as one multiplied by 64 is equal to 64.
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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.
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Finally, differentiating 48 gives us zero, as the differential of any number is equal to zero.
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Therefore, the first derivative of the function π¦ 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.