Given each function, find the equation of the tangent line at each point: y=3(x+1)^2 at the point (2,27)

To find the equation of the tangent line to the function \( y = 3(x + 1)^2 \) at the point \( (2, 27) \), we follow these steps:

### Step 1: Find the derivative of the function
The derivative of a function represents the slope of the tangent line at any point. To find the derivative of \( y = 3(x + 1)^2 \), we apply the chain rule:

\[
\frac{dy}{dx} = 3 \cdot 2(x + 1) \cdot \frac{d}{dx}(x + 1)
\]
Since \( \frac{d}{dx}(x + 1) = 1 \), the derivative simplifies to:
\[
\frac{dy}{dx} = 6(x + 1)
\]

### Step 2: Evaluate the derivative at \( x = 2 \)
The slope of the tangent line at \( x = 2 \) is found by plugging \( x = 2 \) into the derivative:
\[
\frac{dy}{dx} \bigg|_{x = 2} = 6(2 + 1) = 6 \times 3 = 18
\]

Thus, the slope of the tangent line at \( (2, 27) \) is 18.

### Step 3: Use the point-slope form 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 of tangency. In this case, \( m = 18 \) and \( (x_1, y_1) = (2, 27) \).

Substituting these values into the point-slope form:
\[
y - 27 = 18(x - 2)
\]
Simplifying the equation:
\[
y - 27 = 18x - 36
\]
\[
y = 18x - 36 + 27
\]
\[
y = 18x - 9
\]

### Final Answer:
The equation of the tangent line is:
\[
y = 18x - 9
\]

Would you like further explanation or details on any of the steps?

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Here are 5 related questions for further exploration:
1. How do you find the equation of a tangent line for implicit functions?
2. What is the geometric interpretation of a derivative at a point?
3. How do you apply the chain rule to more complex functions?
4. What is the difference between a secant line and a tangent line?
5. How would the equation change if the function was cubic?

**Tip**: When finding the tangent line, always double-check if the point given lies on the curve by substituting it into the original function.

Discover how to find the equation of the tangent line for the function y=3(x+1)^2 at the point (2, 27). This solution involves using the chain rule to compute the derivative and applying the point-slope form for the final equation. Perfect for students in Grades 11-12 and early college.

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