Find the derivative of `f(x)=x^(3)`, by first principle.

Answer

To find the derivative of the function \( f(x) = x^3 \) using the first principle of derivatives, we follow these steps:

### Step 1: Write the definition of the derivative using the first principle.
The derivative \( f'(x) \) can be defined as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

### Step 2: Substitute \( f(x) \) into the derivative formula.
Given \( f(x) = x^3 \), we need to find \( f(x + h) \):
\[ f(x + h) = (x + h)^3 \]
Now, substituting into the derivative formula:
\[ f'(x) = \lim_{h \to 0} \frac{(x + h)^3 - x^3}{h} \]

### Step 3: Expand \( (x + h)^3 \).
Using the binomial expansion:
\[ (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \]
Now substitute this back into the limit:
\[ f'(x) = \lim_{h \to 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} \]

### Step 4: Simplify the expression.
The \( x^3 \) terms cancel out:
\[ f'(x) = \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h} \]
Now, we can factor out \( h \) from the numerator:
\[ f'(x) = \lim_{h \to 0} \frac{h(3x^2 + 3xh + h^2)}{h} \]
Cancelling \( h \) (as long as \( h \neq 0 \)):
\[ f'(x) = \lim_{h \to 0} (3x^2 + 3xh + h^2) \]

### Step 5: Apply the limit as \( h \) approaches 0.
Now we can evaluate the limit:
\[ f'(x) = 3x^2 + 3x(0) + (0)^2 = 3x^2 \]

### Final Answer:
Thus, the derivative of \( f(x) = x^3 \) is:
\[ f'(x) = 3x^2 \]

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