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In this problem, you are asked to find the derivatives \( \frac{dz}{du} \) and \( \frac{dz}{dv} \) using the chain rule. The given relations are:

\[
z = \tan^{-1}\left(\frac{x}{y}\right), \quad x = u \cos v, \quad y = u \sin v.
\]

### Step 1: Rewrite \( z \) in terms of \( u \) and \( v \)
Since \( z = \tan^{-1}\left(\frac{x}{y}\right) \) and \( \frac{x}{y} = \frac{u \cos v}{u \sin v} = \cot v \), we can express \( z \) as:

\[
z = \tan^{-1}(\cot v).
\]

Using the identity \( \tan^{-1}(\cot v) = \frac{\pi}{2} - v \), we now have:

\[
z = \frac{\pi}{2} - v.
\]

### Step 2: Derivatives using the Chain Rule

#### Finding \( \frac{dz}{du} \)
From the expression \( z = \frac{\pi}{2} - v \), we see that \( z \) is independent of \( u \), so:

\[
\frac{dz}{du} = 0.
\]

#### Finding \( \frac{dz}{dv} \)
For \( z = \frac{\pi}{2} - v \), the derivative with respect to \( v \) is:

\[
\frac{dz}{dv} = -1.
\]

### Final Answer:
\[
\frac{dz}{du} = 0, \quad \frac{dz}{dv} = -1.
\]

Would you like further clarification on any of the steps?

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Here are five related questions for further exploration:

1. What would happen if \( x = u \sin v \) and \( y = u \cos v \)?
2. Can you find the second derivative \( \frac{d^2z}{dv^2} \)?
3. How does the identity \( \tan^{-1}(\cot v) = \frac{\pi}{2} - v \) affect the solution?
4. Can we use a different approach to express \( z \) without the identity?
5. How would the derivatives change if \( z = \sin^{-1}\left(\frac{x}{y}\right) \)?

**Tip:** The chain rule is extremely useful when differentiating composite functions, especially when variables are expressed in terms of others. Always look for ways to simplify the function before applying the chain rule.

Find the partial derivatives of z with respect to u and v where z = tan^{-1}(x/y), x = u cos v, and y = u sin v using the chain rule.

Explore the process of finding the derivatives dz/du and dz/dv for the function z = tan^{-1}(x/y) where x and y are defined in terms of u and v. This involves calculus concepts such as the chain rule and trigonometric identities, suitable for college-level students.

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