Also, $$ \left(\frac{d y}{d x}\right)_{x=3}=3.3^{2}=27 $$ Example 2.2: Show that for $$ y=|x|, \frac{d y}{d x} $$ does not exist at $$ x=0 $$

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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.
The derivative of $$ y = |x| $$ does not exist at $$ x = 0 $$ because the left and right derivatives are different.

Also, \( \left(\frac{d y}{d x}\right)_{x=3}=3.3^{2}=27 \) Example 2.2: Show that for \( y=|x|, \frac{d y}{d x} \) does not exist at \( x=0 \)

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