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Find the rate of change of 𝑓 of 𝑥 equals 𝑥 squared minus three 𝑥 plus two at 𝑥 equals five.
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The rate of change can be calculated by differentiating to work out 𝑑𝑦 by 𝑑𝑥 or 𝑓 dash 𝑥.
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In this example, 𝑓 of 𝑥 is equal to 𝑥 squared minus three 𝑥 plus two.
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Differentiating this gives us 𝑓 dash of 𝑥 is equal to two 𝑥 minus three as the differential of 𝑥 squared is two 𝑥, the differential of negative three 𝑥 is negative three, and the differential of two is zero.
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We want to calculate the rate of change at 𝑥 equals five.
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Therefore, 𝑓 dash of five is equal to two multiplied by five minus three.
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Two multiplied by five is 10 minus three equals seven.
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This means that the rate of change of 𝑓 of 𝑥 at 𝑥 equals five is equal to seven.