We are given the equation:

$x^3 f(x) + (f(x))^3 + f(x^3) = 3$ and the initial condition $f(0) = 2$.

Our goal is to find $f'(0)$.

Step 1: Differentiate the equation implicitly

To solve for $f'(0)$, we first differentiate both sides of the equation with respect to $x$.

Left-hand side:

We differentiate term by term.

1. For $x^3 f(x)$, we use the product rule: $\frac{d}{dx} \left( x^3 f(x) \right) = 3x^2 f(x) + x^3 f'(x)$

2. For $(f(x))^3$, we apply the chain rule: $\frac{d}{dx} \left( (f(x))^3 \right) = 3(f(x))^2 f'(x)$

3. For $f(x^3)$, we use the chain rule again: $\frac{d}{dx} \left( f(x^3) \right) = 3x^2 f'(x^3)$

Now, the total derivative of the left-hand side is:

$3x^2 f(x) + x^3 f'(x) + 3(f(x))^2 f'(x) + 3x^2 f'(x^3)$

Right-hand side:

The derivative of the constant 3 is 0.

Step 2: Evaluate at $x = 0$

Now, we substitute $x = 0$ into the differentiated equation. First, recall that $f(0) = 2$, so we substitute that into the equation.

• The term $3x^2 f(x)$ becomes $3(0)^2 f(0) = 0$.
• The term $x^3 f'(x)$ becomes $0$.
• The term $3(f(x))^2 f'(x)$ becomes $3(2)^2 f'(0) = 12 f'(0)$.
• The term $3x^2 f'(x^3)$ becomes $0$.

Thus, at $x = 0$, the equation simplifies to:

$12 f'(0) = 0$

Step 3: Solve for $f'(0)$

From the equation $12 f'(0) = 0$, we conclude that:

$f'(0) = 0$

Final Answer:

$f'(0) = 0$

Would you like more details or have any further questions? Here are some related questions you might find interesting:

1. How do we verify that $f(x) = 2$ is a solution to the original equation?
2. What is the second derivative $f''(0)$?
3. Can we derive a general formula for $f'(x)$ from the given equation?
4. How would we approach solving for $f(x)$ if the equation were different?
5. What is the behavior of the function $f(x)$ near $x = 0$?

Tip: When dealing with implicit differentiation problems like this, always carefully apply the chain rule and product rule, especially when you have functions of $f(x)$ or $f(x^3)$.