Functions of the form \(f(x) = x^n\text{,}\) where \(n = 1, 2, 3, \ldots\text{,}\) are often called power functions. The first two questions below revisit work we did earlier in Chapter 1, and the following questions extend those ideas to higher powers of \(x\text{.}\)
Use the limit definition of the derivative to find \(f'(x)\) for \(f(x) = x^4\text{.}\) (Hint: \((a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\text{.}\) Apply this rule to \((x+h)^4\) within the limit definition.)
Conjecture a formula for the derivative of \(f(x) = x^n\) that holds for any positive integer \(n\text{.}\) That is, given \(f(x) = x^n\) where \(n\) is a positive integer, what do you think is the formula for \(f'(x)\text{?}\)
