https://file.aiquickdraw.com/mathbot/file/172957298986837h5yoy9.jpg

It seems like the image shows a problem where you are asked to find the derivative of a function \( f(x) \) and determine the values of \( x \) where the tangent line is horizontal. Let me break this down:

The function \( f(x) = \frac{x}{6\sqrt{x}} \) is given, and the derivative \( f'(x) \) appears as:

\[
f'(x) = \frac{6 - \frac{x}{2}}{-x + 6^2}
\]

### 1. Finding the Derivative:

The function is of the form \( f(x) = \frac{x}{6\sqrt{x}} \), which simplifies to \( f(x) = \frac{1}{6} \cdot \sqrt{x} \). The derivative of this is found using the power rule:

\[
f(x) = \frac{1}{6} \cdot x^{1/2}
\]
The derivative is:

\[
f'(x) = \frac{1}{6} \cdot \frac{1}{2} x^{-1/2} = \frac{1}{12\sqrt{x}}
\]

### 2. Horizontal Tangent Lines:

The tangent line to the graph is horizontal where \( f'(x) = 0 \). This would require the numerator of the derivative expression to be zero. However, the exact expression shown in the image for \( f'(x) \) seems to differ from the one we calculated here, so it might be an intermediate step or alternate form used in the solution. 

We can check more details on why that \( x \)-value does not satisfy horizontal conditions based on the wrong answer shown (likely due to incorrect solving of the numerator).

Would you like more detailed steps on solving for horizontal tangents?

Here are five questions related to this:

1. How do you apply the chain rule to compute derivatives of functions involving square roots?
2. Why does a tangent line being horizontal imply that the derivative is zero?
3. Can the function \( f(x) = \frac{1}{6} \cdot \sqrt{x} \) have any points where the slope is zero?
4. How can simplifying a function first help in finding its derivative more easily?
5. What happens to the graph of a square root function when \( x \) approaches zero?

**Tip:** Always simplify the function before applying derivative rules. This can help avoid errors!

Find the derivative of the given function: f(x) = x / (6√x) and write all x-values at which the tangent line would be horizontal.

Learn how to find the derivative of f(x) = x / (6√x) using the power and quotient rules. Also, discover how to identify the x-values where the tangent line is horizontal by setting the derivative equal to zero.

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