Differentiate the following w.r.t. x:
|2x - 1|

Answer

To differentiate the function \( f(x) = |2x - 1| \) with respect to \( x \), we need to consider the definition of the absolute value function, which behaves differently based on the sign of the expression inside it.

### Step 1: Identify the critical point
The expression inside the absolute value, \( 2x - 1 \), changes sign at the point where \( 2x - 1 = 0 \). Solving for \( x \):
\[ 2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2} \]
Thus, we have a critical point at \( x = \frac{1}{2} \).

### Step 2: Define the function piecewise
We can define the function \( f(x) \) piecewise based on the critical point:
\[ f(x) = \begin{cases} -(2x - 1) & \text{if } x < \frac{1}{2} \\ 2x - 1 & \text{if } x \geq \frac{1}{2} \end{cases} \]

### Step 3: Differentiate each piece
Now we differentiate each piece of the function separately.

1. For \( x < \frac{1}{2} \):
\[ f(x) = -(2x - 1) = -2x + 1 \]
Differentiating:
\[ f'(x) = -2 \]

2. For \( x \geq \frac{1}{2} \):
\[ f(x) = 2x - 1 \]
Differentiating:
\[ f'(x) = 2 \]

### Step 4: Combine the results
We can summarize the derivative as:
\[ f'(x) = \begin{cases} -2 & \text{if } x < \frac{1}{2} \\ 2 & \text{if } x \geq \frac{1}{2} \end{cases} \]

### Step 5: Conclusion
Thus, the derivative of \( f(x) = |2x - 1| \) with respect to \( x \) is given by the piecewise function above.