Video: Finding the Rate of Change of a Polynomial Function at a Point

Video Transcript

Find the rate of change of 𝑓 of 𝑥 equals five 𝑥 cubed plus 17 when 𝑥 is equal to three.

If we’re given a function 𝑓 of 𝑥, then the rate of change is given by the function 𝑓 dash of 𝑥. This means that we need to differentiate our function for 𝑓 of 𝑥. Our general rule for differentiation states that if 𝑓 of 𝑥 is equal to 𝑎 multiplied by 𝑥 to the power of 𝑛, then 𝑓 dash of 𝑥 is equal to 𝑛 multiplied by 𝑎 multiplied by 𝑥 to the power of 𝑛 minus one. We multiply the power by the coefficient and then subtract one from the power.

In this question, 𝑓 of 𝑥 is equal to five 𝑥 cubed plus 17. We need to differentiate each of these terms individually. To differentiate five 𝑥 cubed, we firstly need to multiply three by five. This gives us 15. Reducing the power by one gives us 𝑥 squared. Therefore, differentiating five 𝑥 cubed gives us 15𝑥 squared. Differentiating any constant gives us zero. Therefore, the differential of 17 is zero. If 𝑓 of 𝑥 is equal to five 𝑥 cubed plus 17 then 𝑓 dash of 𝑥 is equal to 15𝑥 squared. The general rate of change is 15𝑥 squared.

However, we need to work this out when 𝑥 is equal to three. 𝑓 dash of three is equal to 15 multiplied by three squared. Three squared is equal to nine, so we need to multiply 15 by nine. 10 multiplied by nine is equal to 90. Five multiplied by nine is equal to 45. This means that 15 multiplied by nine is equal to 135. The rate of change of the function five 𝑥 cubed plus 17 when 𝑥 equals three is 135.