Answer
The derivative of \( y = |x| \) does not exist at \( x = 0 \) because the left and right derivatives are different.
Solution
Certainly! Let's **demonstrate why the derivative of \( y = |x| \) does not exist at \( x = 0 \)**.
### Understanding the Function \( y = |x| \)
The absolute value function is defined as:
\[
y = |x| =
\begin{cases}
x & \text{if } x \geq 0, \\
-x & \text{if } x < 0.
\end{cases}
\]
This piecewise definition indicates that the function behaves differently on either side of \( x = 0 \).
### Derivative from the Right (\( x \to 0^+ \))
When approaching \( x = 0 \) from the **right** (i.e., \( x > 0 \)):
\[
y = x
\]
The derivative of \( y \) with respect to \( x \) is:
\[
\frac{dy}{dx} = 1
\]
### Derivative from the Left (\( x \to 0^- \))
When approaching \( x = 0 \) from the **left** (i.e., \( x < 0 \)):
\[
y = -x
\]
The derivative of \( y \) with respect to \( x \) is:
\[
\frac{dy}{dx} = -1
\]
### Comparing the One-Sided Derivatives
- **From the Right:** \( \frac{dy}{dx} = 1 \)
- **From the Left:** \( \frac{dy}{dx} = -1 \)
Since these two one-sided derivatives are **not equal**, the overall derivative at \( x = 0 \) **does not exist**.
### Visual Interpretation
Graphically, \( y = |x| \) forms a "V" shape at \( x = 0 \). The slopes of the lines on either side of \( x = 0 \) are slopes of 1 and -1, respectively. At the vertex \( x = 0 \), there's a sharp corner where these two lines meet, creating a point where the function is **not differentiable**.
### Formal Definition Using Limits
To further solidify this, we can use the limit definition of the derivative:
\[
\frac{dy}{dx}\bigg|_{x=0} = \lim_{h \to 0} \frac{|0 + h| - |0|}{h} = \lim_{h \to 0} \frac{|h|}{h}
\]
Evaluating the limit:
- **As \( h \to 0^+ \):**
\[
\lim_{h \to 0^+} \frac{h}{h} = \lim_{h \to 0^+} 1 = 1
\]
- **As \( h \to 0^- \):**
\[
\lim_{h \to 0^-} \frac{-h}{h} = \lim_{h \to 0^-} -1 = -1
\]
Since the left-hand limit (\(-1\)) does not equal the right-hand limit (\(1\)), the overall limit **does not exist**. Therefore, \( \frac{dy}{dx} \) is undefined at \( x = 0 \).
### Conclusion
The derivative \( \frac{dy}{dx} \) for the function \( y = |x| \) does **not exist** at \( x = 0 \) because the slopes from the left and right do not match, resulting in a discontinuity in the derivative at that point.
Reviewed and approved by the UpStudy tutoring team
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