Differentiate each function below. That is, write an equation for the slope function, $f^\prime$.

1. $f\left(x\right) = 6\left(x – 2\right)^{3}$ 

   Hint (a):

   You could expand $f\left(x\right)$ before you differentiate. Or...

   Hint (a):

   Answer (a):

   Strategy 1: $f^\prime \left(x\right) = 18x^2 − 72x + 72$
   Strategy 2: $f^\prime \left(x\right) = 18\left(x − 2\right)^2$

   You know that the derivative of $y = \left(x − 2\right)^3 \text{ is } y^\prime = 3\left(x − 2\right)^2$. What about the vertical stretch of $6$? If the $f\left(x\right)$ is stretched by a factor of $6$, what happens to its slopes/derivative? Are they stretched as well?

2. $f\left(x\right)=2\sin\left(x\right)$ 

   Hint (b):

   What does the $2$ do to a sine graph? How will this affect the slope function?

3. $f\left(x\right) = \left(x + 5\right)\left(2x – 1\right)$ 

   Hint (c):

   Before you differentiate, expand $f\left(x\right).$

4. $f ( x ) = \large\frac { x ^ { 3 } - 6 x ^ { 2 } + 2 x } { x }$ 

   Hint (d):

   Before you differentiate, simplify $f\left(x\right)$. But, don't forget that there was once a $0$ in the denominator... be sure to restrict the domain of the derivative.