Let $f(x) = x^3$.

1. Find the average rate of change from $x = 3$ to $x = 5$.

   Step (a):

   $\frac{f(5)-f(3)}{5-3}$

2. Show that the average rate of change from $x = 4$ to $x = 4 + h$ is $48 + 12h + h^{2}$.

   Step (b):

   $\frac{f(4+h)-f(4)}{(4+h)-4}$

3. Find $f^\prime(4)$, the instantaneous rate of change when $x = 4$.

   Step (c):

   $\lim\limits_{ h \to 0} \frac{f(4+h)-f(4)}{(4+h)-4}$

4. Show that the average rate of change from $x$ to $x + h$ is $3x^{2} + 3xh + h^{2}$.

   Step (d):

   $\frac{f(x+h)-f(x)}{(x+h)-x}$

5. Find $f^\prime\left(x\right)$, the instantaneous rate of change for any value of $x$.

   Step (e):

   $\lim\limits_{ h \to 0}\frac{f(x+h)-f(x)}{(x+h)-x}$