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To find the equation of the line tangent to the curve \( y = 3x \cos x \) at the point \( (\pi, -3\pi) \), we need to follow these steps:

### Step 1: Differentiate \( y = 3x \cos x \)
We start by finding the derivative of \( y \) with respect to \( x \) to get the slope of the tangent line. Using the product rule, where \( u = 3x \) and \( v = \cos x \), we have:

\[
\frac{dy}{dx} = u'v + uv'
\]

Calculating each term:
- \( u = 3x \Rightarrow u' = 3 \)
- \( v = \cos x \Rightarrow v' = -\sin x \)

So,
\[
\frac{dy}{dx} = (3)(\cos x) + (3x)(-\sin x) = 3 \cos x - 3x \sin x
\]

### Step 2: Substitute \( x = \pi \) into \( \frac{dy}{dx} \)
To find the slope of the tangent line at \( x = \pi \), substitute \( x = \pi \) into the derivative:

\[
\frac{dy}{dx} \bigg|_{x=\pi} = 3 \cos(\pi) - 3\pi \sin(\pi)
\]

Since \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \), this simplifies to:

\[
\frac{dy}{dx} \bigg|_{x=\pi} = 3(-1) - 3\pi(0) = -3
\]

So, the slope of the tangent line at \( x = \pi \) is \(-3\).

### Step 3: Write the equation of the tangent line
The point of tangency is \( (\pi, -3\pi) \), and the slope \( m = -3 \). Using the point-slope form of a line \( y - y_1 = m(x - x_1) \):

\[
y - (-3\pi) = -3(x - \pi)
\]

Simplifying, we get:

\[
y + 3\pi = -3x + 3\pi
\]

Subtracting \( 3\pi \) from both sides:

\[
y = -3x
\]

### Final Answer:
The equation of the tangent line at \( (\pi, -3\pi) \) is:

\[
y = -3x
\]

Would you like more details or have any questions?

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Here are five related questions:
1. How do you find the equation of a tangent line for different types of functions?
2. What is the importance of the product rule in calculus?
3. How does the point-slope form of a line help in finding tangent lines?
4. Can tangent lines be vertical, and if so, under what conditions?
5. How would you find the tangent line if given a parametric curve?

**Tip:** Knowing the fundamental differentiation rules (like product, quotient, and chain rules) is essential for tackling calculus problems involving slopes and tangent lines.

Find the equation of the line that is tangent to the curve y = 3x cos x at the point (π, -3π).

Learn how to find the equation of the tangent line to the curve y = 3x cos x at the point (π, -3π). This solution uses calculus concepts such as differentiation and the product rule to determine the slope of the tangent line, followed by applying the point-slope form of a line.

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