Find an equation of the tangent line to the curve, y=x^6(cos(x)) at the point, (pi, -pi^6)

To find the equation of the tangent line to the curve \( y = x^6 \cos(x) \) at the point \( (\pi, -\pi^6) \), we need to follow these steps:

### Step 1: Find the derivative of the function
The equation of the tangent line requires the slope of the curve at the given point, which is obtained from the derivative \( \frac{dy}{dx} \).

The function is \( y = x^6 \cos(x) \). To differentiate this, we will use the **product rule**, which states:
\[
\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)
\]
Let \( f(x) = x^6 \) and \( g(x) = \cos(x) \). Now, differentiate each part:

- \( f'(x) = 6x^5 \)
- \( g'(x) = -\sin(x) \)

Using the product rule:
\[
\frac{dy}{dx} = 6x^5 \cos(x) + x^6 (-\sin(x))
\]
\[
\frac{dy}{dx} = 6x^5 \cos(x) - x^6 \sin(x)
\]

### Step 2: Evaluate the derivative at \( x = \pi \)
To find the slope of the tangent line at the point \( x = \pi \), substitute \( x = \pi \) into the derivative:

\[
\frac{dy}{dx}\bigg|_{x = \pi} = 6\pi^5 \cos(\pi) - \pi^6 \sin(\pi)
\]

We know that:
- \( \cos(\pi) = -1 \)
- \( \sin(\pi) = 0 \)

Thus:
\[
\frac{dy}{dx}\bigg|_{x = \pi} = 6\pi^5 (-1) - \pi^6 (0) = -6\pi^5
\]

So, the slope of the tangent line is \( m = -6\pi^5 \).

### Step 3: Use the point-slope form to find the equation of the tangent line
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is the point on the line.

Using the point \( (\pi, -\pi^6) \) and the slope \( m = -6\pi^5 \), we substitute into the point-slope form:
\[
y - (-\pi^6) = -6\pi^5 (x - \pi)
\]
Simplifying:
\[
y + \pi^6 = -6\pi^5 (x - \pi)
\]

This is the equation of the tangent line to the curve \( y = x^6 \cos(x) \) at the point \( (\pi, -\pi^6) \).

Would you like more details on any specific part of the process or further clarification?

Here are 5 questions for deeper exploration:
1. How do you apply the product rule in differentiation?
2. What is the significance of the point-slope form of a line?
3. Why does \( \cos(\pi) \) equal \( -1 \) and \( \sin(\pi) \) equal \( 0 \)?
4. How do you interpret the derivative in terms of the curve's slope at a point?
5. What are some real-world applications of finding tangent lines?

**Tip**: Remember that the product rule is useful whenever you are differentiating a product of two functions.

Find an equation of the tangent line to the curve, y=x^6(cos(x)) at the point, (pi, -pi^6).

This page provides a detailed solution to finding the tangent line to the curve y = x^6 cos(x) at the point (pi, -pi^6). Learn how to apply the product rule and point-slope form to solve this calculus problem. Suitable for university-level calculus students.

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