Functions of the form $f(x) = x^n$, where $n = 1, 2, 3, \ldots$, are often called power functions.

(a) Use the limit definition of the derivative to find $f'(x)$ for $f(x)=x^2$.
$f'(x) =$

(b) Use the limit definition of the derivative to find $f'(x)$ for $f(x)=x^3$.
$f'(x) =$

(c) Use the limit definition of the derivative to find $f'(x)$ for $f(x)=x^4$. (Hint: $(a + b)^4 = a^4 + 4a^3 b + 6a^2 b^2 + 4ab^3 + b^4$. Apply this rule to $(x + h)^4$ within the limit definition.)
$f'(x) =$

(d) Based on your work in (a), (b), and (c), what do you conjecture is the derivative of $f(x) = x^5$?
Of $f(x) = x^{13}$?

(e) Conjecture a formula for the derivative of $f(x) = x^n$ that holds for any positive integer $n$. That is, given $f(x) = x^n$ where $n$ is a positive integer, what do you think is the formula for $f'(x)$?
$f'(x) =$

You can earn partial credit on this problem.