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To find the equation of the tangent line to the curve \( y = (1 + 4x)^9 \) at the point \( (0, 1) \), we need to:

1. **Differentiate** the given function to find the slope of the tangent line.
2. **Evaluate** the derivative at \( x = 0 \), which gives the slope of the tangent line at the point \( (0, 1) \).
3. Use the **point-slope form** of the equation of a line to find the equation of the tangent.

### Step 1: Differentiate \( y = (1 + 4x)^9 \)

We apply the chain rule to differentiate the function:

\[
\frac{dy}{dx} = 9(1 + 4x)^8 \cdot \frac{d}{dx}(1 + 4x)
\]

\[
\frac{dy}{dx} = 9(1 + 4x)^8 \cdot 4 = 36(1 + 4x)^8
\]

### Step 2: Evaluate the derivative at \( x = 0 \)

Substitute \( x = 0 \) into the derivative:

\[
\frac{dy}{dx} \bigg|_{x=0} = 36(1 + 4(0))^8 = 36(1)^8 = 36
\]

So, the slope of the tangent line at \( (0, 1) \) is \( 36 \).

### Step 3: Use the point-slope form of the equation of a line

The point-slope form is:

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

where \( m = 36 \), and the point is \( (x_1, y_1) = (0, 1) \).

Substitute these values:

\[
y - 1 = 36(x - 0)
\]

Simplify:

\[
y = 36x + 1
\]

### Final Answer:

The equation of the tangent line is:

\[
y = 36x + 1
\]

---

Would you like further details or have any questions? 

Here are 5 related questions to extend this problem:
1. How do you apply the chain rule in more complex functions?
2. What is the significance of the slope of the tangent line at a given point?
3. Can you find the second derivative of the function \( y = (1 + 4x)^9 \)?
4. How would the equation of the tangent line change if the point were different?
5. How can you use derivatives to find local maxima and minima of a function?

**Tip**: Always remember to differentiate carefully when functions involve compositions, applying the chain rule where needed!

Find the equation of the tangent line to the curve y = (1 + 4x)^9 at the point (0, 1).

Learn how to find the equation of the tangent line to the curve y = (1 + 4x)^9 at the point (0, 1). This problem applies differentiation using the chain rule to compute the slope and uses the point-slope form to find the equation. Suitable for high school students in Grades 10-12.

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