Find, from first principles, the derivative of the following w.r.t. x :
`x^(3)`

Answer

To find the derivative of the function \( f(x) = x^3 \) from first principles, we will follow the definition of the derivative. The derivative of a function \( f(x) \) at a point \( x \) is given by:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

### Step-by-step Solution:

1. **Identify the function**:
We have \( f(x) = x^3 \).

2. **Set up the limit**:
We need to calculate \( f'(x) \):
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

3. **Calculate \( f(x + h) \)**:
Substitute \( x + h \) into the function:
\[ f(x + h) = (x + h)^3 \]

4. **Expand \( (x + h)^3 \)**:
Using the binomial expansion:
\[ (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \]

5. **Substitute into the limit**:
Now substitute \( f(x + h) \) and \( f(x) \) into the limit:
\[ f'(x) = \lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h} \]

6. **Simplify the expression**:
The \( x^3 \) terms cancel out:
\[ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} \]

7. **Factor out \( h \)**:
We can factor \( h \) out of the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2)}{h} \]

8. **Cancel \( h \)**:
The \( h \) in the numerator and denominator cancels out (as long as \( h \neq 0 \)):
\[ f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) \]

9. **Take the limit as \( h \to 0 \)**:
Now, substitute \( h = 0 \):
\[ f'(x) = 3x^2 + 3x(0) + (0)^2 = 3x^2 \]

10. **Final result**:
Thus, the derivative of \( f(x) = x^3 \) is:
\[ f'(x) = 3x^2 \]