The problem in the image seems to involve calculus. Here's a breakdown of what I see:

The question appears to be asking for the third derivative $\frac{d^3y}{dx^3}$. The given conditions are:

1. $y = f(f(x))$ — this is a composite function involving $f$ applied to $f(x)$.
2. There’s an expression provided: $f(x + h) - f(x) = 5x^3 h$ This resembles the form of a difference quotient, which provides clues about the derivative of $f(x)$.

Solution:

1. Interpretation of the given equation:

   • From the difference quotient: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \lim_{h \to 0} \frac{5x^3h}{h} = 5x^3$ So, the first derivative of $f(x)$ is: $f'(x) = 5x^3$

2. Finding the second derivative of $f$: $f''(x) = \frac{d}{dx}(5x^3) = 15x^2$

3. Finding the third derivative of $f$: $f'''(x) = \frac{d}{dx}(15x^2) = 30x$

4. Handling the composite function $y = f(f(x))$:

   • Use the chain rule to differentiate it step by step.

   • First derivative: $\frac{dy}{dx} = f'(f(x)) \cdot f'(x)$

   • Second derivative (using product and chain rules): $\frac{d^2y}{dx^2} = f''(f(x)) \cdot (f'(x))^2 + f'(f(x)) \cdot f''(x)$

   • Third derivative (this will involve more terms): $\frac{d^3y}{dx^3} = f'''(f(x)) \cdot (f'(x))^3 + 3 f''(f(x)) \cdot f'(x) \cdot f''(x) + f'(f(x)) \cdot f'''(x)$

5. Substitute the known derivatives:

   • $f'(x) = 5x^3$
   • $f''(x) = 15x^2$
   • $f'''(x) = 30x$

   Substitute these values into the expression for the third derivative.

If you need further steps or clarifications on a specific part, let me know! Now here are some related questions to expand your understanding:

Related Questions:

1. What is the significance of higher-order derivatives in physics or economics?
2. How does the chain rule work for nested functions like $f(f(x))$?
3. How can the difference quotient be used to derive the first derivative directly?
4. What is the meaning of $f'(x) = 5x^3$ in terms of the slope of the function?
5. How do product and chain rules combine when differentiating complex expressions?

Tip:

Always break down composite functions systematically by applying the chain rule step-by-step to avoid confusion during differentiation.