Given `f(x)=2x^(3)`, find `f^(')(x)` by delta method.

Answer

To find the derivative \( f'(x) \) of the function \( f(x) = 2x^3 \) using the delta method, we will follow these steps:

### Step 1: Define the function and the increment
We start with the function:
\[ f(x) = 2x^3 \]
We will also consider \( f(x + h) \) where \( h \) is a small increment:
\[ f(x + h) = 2(x + h)^3 \]

### Step 2: Expand \( f(x + h) \)
We need to expand \( f(x + h) \):
\[ f(x + h) = 2(x + h)^3 = 2(x^3 + 3x^2h + 3xh^2 + h^3) \]
This simplifies to:
\[ f(x + h) = 2x^3 + 6x^2h + 6xh^2 + 2h^3 \]

### Step 3: Set up the difference quotient
The difference quotient is given by:
\[ \frac{f(x + h) - f(x)}{h} \]
Substituting the expressions we have:
\[ \frac{(2x^3 + 6x^2h + 6xh^2 + 2h^3) - 2x^3}{h} \]
This simplifies to:
\[ \frac{6x^2h + 6xh^2 + 2h^3}{h} \]

### Step 4: Simplify the difference quotient
Now we can simplify the expression:
\[ \frac{6x^2h + 6xh^2 + 2h^3}{h} = 6x^2 + 6xh + 2h^2 \]

### Step 5: Take the limit as \( h \to 0 \)
Now we take the limit as \( h \) approaches 0:
\[ \lim_{h \to 0} (6x^2 + 6xh + 2h^2) = 6x^2 + 0 + 0 = 6x^2 \]

### Conclusion
Thus, we find that the derivative of the function \( f(x) = 2x^3 \) is:
\[ f'(x) = 6x^2 \]