CPM Homework Help : CALC Problem 4-20

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4-20.

Differentiate each function below. That is, find its slope function, $f^\prime(x)$ .

$f(x) = 6(x-2)^3$
Hint 1(a):
You could expand $f(x)$ before you differentiate. Or...
Hint 2(a):
You know that the derivative of $y = (x − 2)^³$ is $y^\prime= 3(x-2)^²$ . What about the vertical stretch of $6$ ? If the $f(x)$ is stretched by a factor of $6$ , what happens to its slopes/derivative? Are they stretched as well?
Answer (a):
Strategy 1: $f^\prime(x) = 18x^²-72x + 72$
Strategy 2: $f^\prime(x) = 18(x-2)^²$

$f(x) = 2\operatorname{ sin }x$
Hint (b):
What does the $2$ do to a sine graph? How will this affect the slope function?

$f(x) = (x + 5)(2x − 1)$
Hint (c):
Before you differentiate, expand $f(x)$ .

$f ( x ) = \frac { x ^ { 3 } - 6 x ^ { 2 } + 2 x } { x }$
Hint (d):
Before you differentiate, simplify $f(x)$ . But, don't forget that there was once a $0$ in the denominator... be sure to restrict the domain of the derivative.

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