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To find the equation of the tangent line to the function \( f(x) = -3x^3 - 2x^2 - x + 3 \) at the point \((-1, 5)\), we will follow these steps:

1. **Find the derivative** of \( f(x) \), \( f'(x) \), which will give the slope of the tangent line at any point \(x\).
2. **Evaluate the derivative at \( x = -1 \)** to find the slope of the tangent line at the point \( (-1, 5) \).
3. **Use the point-slope form** of the equation of a line: \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \( (x_1, y_1) \) is the point, and then rewrite it in the slope-intercept form \( y = mx + b \).

### Step 1: Find the derivative \( f'(x) \)
Given \( f(x) = -3x^3 - 2x^2 - x + 3 \), apply the power rule to differentiate each term:
\[
f'(x) = \frac{d}{dx} \left( -3x^3 \right) + \frac{d}{dx} \left( -2x^2 \right) + \frac{d}{dx} \left( -x \right) + \frac{d}{dx} \left( 3 \right)
\]
\[
f'(x) = -9x^2 - 4x - 1
\]

### Step 2: Evaluate the derivative at \( x = -1 \)
To find the slope of the tangent line at the point \( (-1, 5) \), substitute \( x = -1 \) into \( f'(x) \):
\[
f'(-1) = -9(-1)^2 - 4(-1) - 1
\]
\[
f'(-1) = -9 + 4 - 1 = -6
\]
Thus, the slope of the tangent line is \( m = -6 \).

### Step 3: Use the point-slope form
The equation of the tangent line can be written using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) = (-1, 5) \) and \( m = -6 \):
\[
y - 5 = -6(x + 1)
\]
Simplify the equation:
\[
y - 5 = -6x - 6
\]
\[
y = -6x - 1
\]

### Final Answer
The equation of the tangent line is:
\[
y = -6x - 1
\]

Would you like more details on any of these steps, or do you have any further questions? Here are 5 related questions you might find helpful:

1. How do you differentiate higher-order polynomials?
2. What is the geometric interpretation of a tangent line?
3. How does the slope of the tangent relate to instantaneous rates of change?
4. Can the point-slope form be used for any line, not just tangent lines?
5. What is the connection between derivatives and the behavior of functions (e.g., increasing or decreasing)?

**Tip:** Always check that the point given (in this case, \( (-1, 5) \)) satisfies the original function \( f(x) \) to ensure it lies on the curve before calculating the tangent.

Find the equation of the tangent line to the function f(x) = -3x^3 - 2x^2 - x + 3 at the point (-1, 5).

Learn how to find the equation of the tangent line to the function f(x) = -3x^3 - 2x^2 - x + 3 at the point (-1, 5). This step-by-step solution covers calculus concepts including derivatives and the tangent line equation. Suitable for Grade 12 or College-level students.

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