Show that `f(x)=x^2` is differentiable at `x=1` and find `f^(prime)(1)` .

Explanation

To show that the function \( f(x) = x^2 \) is differentiable at \( x = 1 \) and to find \( f'(1) \), we will calculate the left-hand derivative and the right-hand derivative at that point.

### Step 1: Calculate the Left-Hand Derivative (LHD)

The left-hand derivative at \( x = 1 \) is given by the limit:

\[
\text{LHD} = \lim_{h \to 0} \frac{f(1 - h) - f(1)}{(1 - h) - 1}
...