The expression you're dealing with is:

$\frac{f(a+h) - f(a)}{h}$

where $f(x) = 3x^2 + 3$.

Step 1: Find $f(a+h)$

Substitute $a + h$ for $x$ in the function $f(x)$:

$f(a+h) = 3(a+h)^2 + 3$

Now expand $(a+h)^2$:

$f(a+h) = 3(a^2 + 2ah + h^2) + 3$

Simplifying:

$f(a+h) = 3a^2 + 6ah + 3h^2 + 3$

Step 2: Calculate $f(a+h) - f(a)$

Next, subtract $f(a)$ from $f(a+h)$:

$f(a) = 3a^2 + 3$

So:

$f(a+h) - f(a) = (3a^2 + 6ah + 3h^2 + 3) - (3a^2 + 3)$

Simplify:

$f(a+h) - f(a) = 6ah + 3h^2$

Step 3: Divide by $h$

Now, divide the expression by $h$:

$\frac{f(a+h) - f(a)}{h} = \frac{6ah + 3h^2}{h}$

Factor out $h$ from the numerator:

$\frac{f(a+h) - f(a)}{h} = \frac{h(6a + 3h)}{h}$

Since $h \neq 0$, the $h$'s cancel out:

$\frac{f(a+h) - f(a)}{h} = 6a + 3h$

Final Result

$\frac{f(a+h) - f(a)}{h} = 6a + 3h$

This is your final expression.

Let me know if you'd like more details on this process or if you have any questions!

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Related Questions:

1. What is the derivative of $f(x) = 3x^2 + 3$?
2. How does the limit of $\frac{f(a+h) - f(a)}{h}$ as $h \to 0$ relate to the derivative?
3. What would happen if we were to use a different function, like $f(x) = x^3 + 5$, for the same expression?
4. How do you simplify expressions like $\frac{f(a+h) - f(a)}{h}$ when $f(x)$ has higher degree terms?
5. What role does the concept of a limit play in finding derivatives?

Tip: The expression $\frac{f(a+h) - f(a)}{h}$ is key to understanding the definition of the derivative, and the limit as $h \to 0$ reveals the slope of the tangent line to the curve at a point.