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Let's start by solving the problem step by step.

### 1. Find \( f'(-1) \) using the definition of the derivative:

The function is \( f(x) = 3x^2 + 3x + 3 \). According to the definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

For \( f'(-1) \), we substitute \( x = -1 \):

\[
f'(-1) = \lim_{h \to 0} \frac{f(-1+h) - f(-1)}{h}
\]

First, calculate \( f(-1) \):

\[
f(-1) = 3(-1)^2 + 3(-1) + 3 = 3(1) - 3 + 3 = 3
\]

Next, calculate \( f(-1+h) \):

\[
f(-1+h) = 3(-1+h)^2 + 3(-1+h) + 3
\]

Expand \( (-1+h)^2 = 1 - 2h + h^2 \):

\[
f(-1+h) = 3(1 - 2h + h^2) + 3(-1+h) + 3 = 3(1 - 2h + h^2) - 3 + 3h + 3
\]
\[
= 3 - 6h + 3h^2 - 3 + 3h + 3 = 3h^2 - 3h + 3
\]

Now calculate the difference:

\[
f(-1+h) - f(-1) = (3h^2 - 3h + 3) - 3 = 3h^2 - 3h
\]

Now apply the limit:

\[
f'(-1) = \lim_{h \to 0} \frac{3h^2 - 3h}{h} = \lim_{h \to 0} (3h - 3) = -3
\]

Thus, \( f'(-1) = -3 \).

### 2. Equation of the tangent line at \( (-1, 3) \):

The equation of the tangent line can be written in point-slope form:

\[
y - y_1 = m(x - x_1)
\]

Where \( m = f'(-1) = -3 \), and the point is \( (-1, 3) \). Substituting:

\[
y - 3 = -3(x + 1)
\]

Simplifying:

\[
y - 3 = -3x - 3 \quad \Rightarrow \quad y = -3x
\]

### Final Answer:
- The derivative \( f'(-1) = -3 \).
- The equation of the tangent line is \( y = -3x \).

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:
1. How do you find the derivative of a function using the power rule?
2. What is the geometric interpretation of a derivative at a point?
3. How does the definition of the derivative relate to limits?
4. How can we apply derivatives to find the equation of a tangent line at any point?
5. How do higher-order derivatives relate to the shape of a graph?

**Tip**: The derivative of a quadratic function always gives a linear function as its result.

If f(x) = 3x^2 + 3x + 3, find f'(-1) using the definition of derivative. f'(-1) is the limit as x → -1 of the expression. Use this to find the equation of the tangent line to the parabola y = 3x^2 + 3x + 3 at the point (-1, 3).

Learn how to find the derivative f'(-1) of f(x) = 3x^2 + 3x + 3 using the limit definition of a derivative and how to calculate the equation of the tangent line to the parabola at the point (-1, 3).

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