Differentiate the following functions with respect to `x` from definition.
(i) `x^2`

Answer

To differentiate the function \( f(x) = x^2 \) with respect to \( x \) using the definition of the derivative, we will follow these steps:

### Step-by-Step Solution:

1. **Definition of the Derivative**:
The derivative of a function \( f(x) \) is defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

2. **Substituting the Function**:
For our function \( f(x) = x^2 \), we need to find \( f(x+h) \):
\[ f(x+h) = (x+h)^2 \]
Therefore, we can substitute into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} \]

3. **Expanding \( (x+h)^2 \)**:
Using the identity \( (a+b)^2 = a^2 + 2ab + b^2 \):
\[ (x+h)^2 = x^2 + 2xh + h^2 \]
Now substituting this back into our limit:
\[ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \]

4. **Simplifying the Expression**:
The \( x^2 \) terms cancel out:
\[ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} \]

5. **Factoring out \( h \)**:
We can factor \( h \) out of the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} \]
Since \( h \neq 0 \) in the limit, we can cancel \( h \):
\[ f'(x) = \lim_{h \to 0} (2x + h) \]

6. **Taking the Limit**:
Now, as \( h \) approaches 0:
\[ f'(x) = 2x + 0 = 2x \]

### Final Result:
Thus, the derivative of \( f(x) = x^2 \) is:
\[ f'(x) = 2x \]