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.