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