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We are asked to find the equation of the tangent line to the graph of the function:

\[
f(x) = \sin\left(9 \sqrt{x}\right)
\]

at the point \(\left(\pi^2, 0\right)\).

### Step 1: Find the derivative of \( f(x) \)

To get the equation of the tangent line, we need to compute the derivative \( f'(x) \), which gives us the slope of the tangent line at a given point.

Given \( f(x) = \sin(9 \sqrt{x}) \), we will apply the chain rule for differentiation:

- The outer function is \( \sin(u) \), where \( u = 9 \sqrt{x} \).
- The derivative of \( \sin(u) \) is \( \cos(u) \cdot u' \).

Now, we need to find the derivative of \( u = 9 \sqrt{x} = 9 x^{1/2} \). Using the power rule, the derivative is:

\[
u' = 9 \cdot \frac{1}{2} x^{-1/2} = \frac{9}{2 \sqrt{x}}
\]

Now, applying the chain rule:

\[
f'(x) = \cos(9 \sqrt{x}) \cdot \frac{9}{2 \sqrt{x}}
\]

### Step 2: Compute the slope at \( x = \pi^2 \)

To find the slope of the tangent line at \( x = \pi^2 \), we substitute \( x = \pi^2 \) into the derivative \( f'(x) \):

\[
f'(\pi^2) = \cos(9 \sqrt{\pi^2}) \cdot \frac{9}{2 \sqrt{\pi^2}}
\]

Since \( \sqrt{\pi^2} = \pi \), we simplify the expression:

\[
f'(\pi^2) = \cos(9 \pi) \cdot \frac{9}{2 \pi}
\]

We know that \( \cos(9 \pi) = -1 \) because \( \cos(\theta) \) has a period of \( 2\pi \) and \( 9\pi \) corresponds to an odd multiple of \( \pi \). Therefore:

\[
f'(\pi^2) = -1 \cdot \frac{9}{2 \pi} = -\frac{9}{2 \pi}
\]

So, the slope of the tangent line at \( x = \pi^2 \) is \( -\frac{9}{2 \pi} \).

### Step 3: Equation of the tangent line

The general form of the equation of a line is:

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

where \( m \) is the slope, and \( (x_1, y_1) \) is the point of tangency. In our case, \( (x_1, y_1) = (\pi^2, 0) \), and the slope \( m = -\frac{9}{2 \pi} \). Substituting these values:

\[
y - 0 = -\frac{9}{2 \pi} (x - \pi^2)
\]

Simplifying:

\[
y = -\frac{9}{2 \pi} (x - \pi^2)
\]

Thus, the equation of the tangent line is:

\[
y = -\frac{9}{2 \pi} (x - \pi^2)
\]

---

Would you like more details on any part of the solution?

Here are some related questions:

1. How is the chain rule used in differentiation?
2. What is the geometric significance of a tangent line?
3. How do you find the cosine of an angle like \( 9\pi \)?
4. Why is the derivative important for finding the slope of a tangent line?
5. How does the power rule apply to square root functions?

**Tip:** When differentiating composite functions, always identify the inner and outer functions clearly to apply the chain rule properly.

Find the equation of the tangent line to the graph of the function f(x) = sin(9√x) at the point (π^2, 0).

Learn how to find the equation of the tangent line to the function f(x) = sin(9√x) at the point (π^2, 0) using differentiation and the chain rule. This problem involves finding the derivative and applying it to determine the slope of the tangent line, suitable for students in Grades 11-12.

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