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To find the equation of the tangent line to \( y = f(x) = \tan(\pi x) \) at \( x = 1 \), we need to:

1. Find the derivative \( f'(x) \) to determine the slope of the tangent line at \( x = 1 \).
2. Calculate \( f(1) \) to get the point on the function at \( x = 1 \).
3. Use the point-slope form of the line equation.

Let's go through these steps.

### Step 1: Find \( f'(x) \)
Given \( f(x) = \tan(\pi x) \), we can find \( f'(x) \) using the chain rule.

\[
f'(x) = \frac{d}{dx} \left( \tan(\pi x) \right) = \sec^2(\pi x) \cdot \pi = \pi \sec^2(\pi x)
\]

### Step 2: Calculate \( f'(1) \)
Now, substitute \( x = 1 \) into \( f'(x) \):

\[
f'(1) = \pi \sec^2(\pi \cdot 1) = \pi \sec^2(\pi) = \pi \cdot 1 = \pi
\]

So, the slope of the tangent line at \( x = 1 \) is \( \pi \).

### Step 3: Calculate \( f(1) \)
Now, evaluate \( f(1) \):

\[
f(1) = \tan(\pi \cdot 1) = \tan(\pi) = 0
\]

Thus, the point on the curve at \( x = 1 \) is \( (1, 0) \).

### Step 4: Write the Equation of the Tangent Line
Using the point-slope form of a line, \( y - y_1 = m(x - x_1) \), with \( m = \pi \), \( x_1 = 1 \), and \( y_1 = 0 \):

\[
y - 0 = \pi(x - 1)
\]

Simplifying, we get:

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

### Final Answer
The equation of the tangent line is:
\[
\boxed{y = \pi x - \pi}
\]

### Conclusion
The correct answer is **(a) \( y = \pi x - \pi \)**.

Would you like more details, or do you have any questions about these steps?

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Here are some follow-up questions to extend this problem:

1. How would the tangent line change if the function were \( \tan(2\pi x) \) instead?
2. What would the tangent line be at \( x = 0.5 \) for the same function \( f(x) = \tan(\pi x) \)?
3. How does the periodic nature of \( \tan(\pi x) \) affect the behavior of the tangent lines?
4. Can we find points where the tangent line to \( f(x) = \tan(\pi x) \) has a zero slope?
5. What happens to the tangent line as \( x \) approaches points where \( \tan(\pi x) \) is undefined?

**Tip:** Remember, the derivative \( \sec^2(x) \) is crucial for functions involving \( \tan(x) \) when differentiating.

The equation of the tangent line of the function y = f(x) = tan(pi*x) at x = 1 is: a) y = pi*x - pi, b) y = pi*x + pi, c) y = -pi*x - pi, d) y = -pi*x + pi.

Discover how to find the tangent line of the function y = tan(pi*x) at x = 1. This problem involves differentiation using the chain rule and determining the slope with trigonometric functions. Learn how to use point-slope form to arrive at the solution, suitable for students in Grades 11-12.

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