Let's solve the given problem step-by-step.

Step 1: Start with the definition of the derivative

The student is using the limit definition of the derivative: $f'(x) = \lim_{h \to 0} \frac{(x + h)^3 - x^3}{h}$

Step 2: Expand $(x + h)^3$ using the Binomial Theorem

The expansion of $(x + h)^3$ is: $(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$

Step 3: Substitute and simplify the expression

Substituting this back into the limit expression, we have: $f'(x) = \lim_{h \to 0} \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h}$ $= \lim_{h \to 0} \frac{3x^2h + 3xh^2 + h^3}{h}$ $= \lim_{h \to 0} (3x^2 + 3xh + h^2)$

Step 4: Identify the expression in the given form

The expression $f'(x) = \lim_{h \to 0} (Ax^B + Cx^D h^E + h^F)$ corresponds to our simplified expression $3x^2 + 3xh + h^2$.

By comparing terms, we have:

• $Ax^B = 3x^2$, so $A = 3$ and $B = 2$,
• $Cx^D h^E = 3xh$, so $C = 3$, $D = 1$, and $E = 1$,
• $h^F = h^2$, so $F = 2$.

Therefore, the value of $C$ is:

$C = 3$

Would you like further details or explanations?

Here are 5 related questions to expand your understanding:

1. How would the derivative change if $f(x) = x^4$ using the limit definition?
2. What does the Binomial Theorem state, and how is it applied to polynomials?
3. How is the limit definition of the derivative connected to instantaneous rates of change?
4. Can you find the derivative of a polynomial function without using the limit definition?
5. What happens if you apply the same derivative process to $f(x) = x^n$ for any integer $n$?

Tip:

Always double-check the expansion step in the binomial theorem, as it's crucial for correctly simplifying derivatives using the limit definition.