WEBVTT
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Find the rate of change of π of π₯ equals five π₯ cubed plus 17 when π₯ is equal to three.
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If weβre given a function π of π₯, then the rate of change is given by the function π dash of π₯.
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This means that we need to differentiate our function for π of π₯.
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Our general rule for differentiation states that if π of π₯ is equal to π multiplied by π₯ to the power of π, then π dash of π₯ is equal to π multiplied by π multiplied by π₯ to the power of π minus one.
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We multiply the power by the coefficient and then subtract one from the power.
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In this question, π of π₯ is equal to five π₯ cubed plus 17.
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We need to differentiate each of these terms individually.
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To differentiate five π₯ cubed, we firstly need to multiply three by five.
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This gives us 15.
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Reducing the power by one gives us π₯ squared.
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Therefore, differentiating five π₯ cubed gives us 15π₯ squared.
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Differentiating any constant gives us zero.
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Therefore, the differential of 17 is zero.
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If π of π₯ is equal to five π₯ cubed plus 17 then π dash of π₯ is equal to 15π₯ squared.
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The general rate of change is 15π₯ squared.
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However, we need to work this out when π₯ is equal to three.
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π dash of three is equal to 15 multiplied by three squared.
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Three squared is equal to nine, so we need to multiply 15 by nine.
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10 multiplied by nine is equal to 90.
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Five multiplied by nine is equal to 45.
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This means that 15 multiplied by nine is equal to 135.
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The rate of change of the function five π₯ cubed plus 17 when π₯ equals three is 135.