Consider the following function. (If an answer does not exist, enter DNE.)
f(x) = |x − 4|
Find the derivative from the left at x = 4.

Find the derivative from the right at x = 4.

Is the function differentiable at x = 4?

We are given the function \( f(x) = |x - 4| \), and we are tasked with finding the derivative from the left and from the right at \( x = 4 \), and determining whether the function is differentiable at \( x = 4 \).

### 1. **Derivative from the left at \( x = 4 \):**

First, let's rewrite the absolute value function piecewise:

\[
f(x) =
\begin{cases} 
4 - x & \text{if } x < 4 \\
x - 4 & \text{if } x \geq 4 
\end{cases}
\]

To find the derivative from the left, we consider \( f(x) = 4 - x \) for \( x < 4 \). The derivative of \( f(x) = 4 - x \) is:

\[
f'(x) = -1 \quad \text{for} \quad x < 4
\]

Thus, the derivative from the left at \( x = 4 \) is:

\[
f'_-(4) = -1
\]

### 2. **Derivative from the right at \( x = 4 \):**

To find the derivative from the right, we consider \( f(x) = x - 4 \) for \( x \geq 4 \). The derivative of \( f(x) = x - 4 \) is:

\[
f'(x) = 1 \quad \text{for} \quad x > 4
\]

Thus, the derivative from the right at \( x = 4 \) is:

\[
f'_+(4) = 1
\]

### 3. **Is the function differentiable at \( x = 4 \)?**

For a function to be differentiable at a point, the derivative from the left and the derivative from the right must be equal. In this case:

\[
f'_-(4) = -1 \quad \text{and} \quad f'_+(4) = 1
\]

Since \( -1 \neq 1 \), the derivative from the left and the derivative from the right are not equal. Therefore, the function is **not differentiable** at \( x = 4 \).

### Summary:
- The derivative from the left at \( x = 4 \) is \( -1 \).
- The derivative from the right at \( x = 4 \) is \( 1 \).
- The function is not differentiable at \( x = 4 \).

Would you like more details or have any questions?

### Related questions:
1. How do we interpret differentiability geometrically?
2. What happens to the tangent line at a point where a function is not differentiable?
3. Can a function be continuous but not differentiable? Provide examples.
4. What other functions are not differentiable at certain points?
5. How does the absolute value function differ from other common functions in terms of differentiability?

**Tip:** A function's differentiability implies continuity, but the converse is not always true.

Consider the following function. (If an answer does not exist, enter DNE.) f(x) = |x − 4|. Find the derivative from the left at x = 4. Find the derivative from the right at x = 4. Is the function differentiable at x = 4?

Explore the process of finding the derivative from the left and right of the absolute value function f(x) = |x − 4| at x = 4. Learn why the function is not differentiable at this point. Suitable for students in Grades 11-12 studying calculus.

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