Determine the slope function​ S(x) for the following function. 
f left parenthesis x right parenthesis equals StartAbsoluteValue 5 x EndAbsoluteValuef(x)=5x
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Part 1
Select the correct choice below and fill in the answer​ box(es) to complete your choice.
​(Use integers or decimals for any numbers in the​ expression.)
A.
The slope function is the piecewise function Upper S left parenthesis x right parenthesis equals left brace Start 2 By 2 Matrix 1st Row 1st Column nothing 2nd Column if x greater than nothing 2nd Row 1st Column nothing 2nd Column if x less than nothing EndMatrixS(x)= enter your response here	if x>enter your response here	enter your response here	if x<enter your response here .
B.
The slope function is the function ​S(x)equals=enter your response here. 

The given function is \( f(x) = |5x| \). To determine the slope function \( S(x) \), we need to find its derivative. Since the absolute value function involves a piecewise definition, we will break it down into two cases:

1. **For \( x > 0 \):**  
   \( |5x| = 5x \), so \( f(x) = 5x \). The derivative of \( f(x) = 5x \) is \( S(x) = 5 \).

2. **For \( x < 0 \):**  
   \( |5x| = -5x \), so \( f(x) = -5x \). The derivative of \( f(x) = -5x \) is \( S(x) = -5 \).

At \( x = 0 \), the derivative is undefined because of the sharp corner in the absolute value function.

Thus, the slope function \( S(x) \) is a piecewise function:

\[
S(x) =
\begin{cases}
5, & \text{if } x > 0, \\
-5, & \text{if } x < 0.
\end{cases}
\]

### The correct choice is A:
The slope function is the piecewise function:
\[
S(x) = 
\begin{cases}
5 & \text{if } x > 0, \\
-5 & \text{if } x < 0.
\end{cases}
\]

Would you like more details on how to handle absolute value functions in calculus?

Here are 5 related questions to expand your understanding:
1. How do you compute the derivative of \( |x| \)?
2. What is the derivative of \( |x^2 - 4| \)?
3. Can an absolute value function be differentiable at \( x = 0 \)?
4. How does the derivative behave at points where the function has sharp corners?
5. What is the geometric interpretation of the derivative for piecewise linear functions?

**Tip:** When dealing with absolute value functions, always break the function into cases for positive and negative values of the variable.

Determine the slope function S(x) for the following function: f(x) = |5x|. Select the correct choice below and fill in the answer box(es) to complete your choice.

Understand how to determine the slope function for the absolute value function f(x) = |5x| using calculus. This problem involves piecewise functions and finding derivatives for different cases, covering key concepts in calculus for Grades 11-12.

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