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