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Find the rate of change of five π₯ cubed minus 18 with respect to π₯ when π₯ is equal to two.
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The rate of change of a function or expression is just its derivative.
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The derivative of our function π at some number π π prime of π is equal to the limit of π of π plus β minus π of π all over β as β approaches zero.
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If we let π of π₯ equal five π₯ cubed minus 18, then weβre looking for the rate of change of π of π₯ with respect to π₯ when π₯ is equal to two.
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And this is the derivative of π of π₯ at π₯ equals two.
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And we get the derivative of π at the number two by substituting two for π in the definition above.
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Now, we have to evaluate π of two plus β and π of two.
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What is π of two plus β?
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Well, itβs what you get by substituting two plus β for π₯ in the expression for π of π₯.
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In other words, itβs five times two plus β cubed minus 18.
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How about π of two?
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Well, this is five times two cubed minus 18.
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Now, we can simplify the numerator and we can start by binomial expanding five times two plus β cubed.
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We get five times β cubed plus six β squared plus 12β plus eight.
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And we can distribute that five over the terms in the parentheses and we get the following.
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The other terms combine.
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So minus 18 minus five times two cubed minus 18 becomes minus 40.
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And we can see that the two constant terms cancel, leaving us with just five β cubed plus 30β squared plus 60β in the numerator.
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And if we put in the common denominator β as well, we notice that the terms in the numerator have a common factor of this denominator β and so we can cancel.
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Cancelling the βs, we get the limit of five β squared plus 30β plus 60 as β approaches zero.
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And this is a limit that we can evaluate using direct substitution, just substituting zero for β.
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Directly substituting then, we get five times zero squared plus 30 times zero plus 60, which is of course just 60.
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The value of this limit and hence the rate of change of five π₯ cubed minus 18 with respect to the π₯ when π₯ is equal to two is 60.