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The equation given is:

\[
\theta^{-\frac{7}{9}} - r^{-\frac{7}{9}} = 3
\]

We are tasked with finding \(\frac{dr}{d\theta}\), which involves differentiating \(r\) with respect to \(\theta\).

### Steps:

1. Differentiate both sides of the equation with respect to \(\theta\).
   
   \[
   \frac{d}{d\theta}\left( \theta^{-\frac{7}{9}} - r^{-\frac{7}{9}} \right) = \frac{d}{d\theta}(3)
   \]

   The derivative of the constant on the right-hand side is zero.

   \[
   \frac{d}{d\theta}\left( \theta^{-\frac{7}{9}} \right) - \frac{d}{d\theta}\left( r^{-\frac{7}{9}} \right) = 0
   \]

2. Compute the derivative of \(\theta^{-\frac{7}{9}}\):
   
   \[
   \frac{d}{d\theta}\left( \theta^{-\frac{7}{9}} \right) = -\frac{7}{9}\theta^{-\frac{16}{9}}
   \]

3. Use the chain rule to differentiate \(r^{-\frac{7}{9}}\) with respect to \(\theta\), treating \(r\) as a function of \(\theta\):
   
   \[
   \frac{d}{d\theta}\left( r^{-\frac{7}{9}} \right) = -\frac{7}{9}r^{-\frac{16}{9}}\frac{dr}{d\theta}
   \]

4. Substituting these into the equation:

   \[
   -\frac{7}{9}\theta^{-\frac{16}{9}} - \left(-\frac{7}{9}r^{-\frac{16}{9}}\frac{dr}{d\theta}\right) = 0
   \]

   Simplifying:

   \[
   -\frac{7}{9}\theta^{-\frac{16}{9}} + \frac{7}{9}r^{-\frac{16}{9}}\frac{dr}{d\theta} = 0
   \]

5. Solve for \(\frac{dr}{d\theta}\):

   \[
   \frac{7}{9}r^{-\frac{16}{9}}\frac{dr}{d\theta} = \frac{7}{9}\theta^{-\frac{16}{9}}
   \]

   Cancel the \(\frac{7}{9}\) terms:

   \[
   r^{-\frac{16}{9}}\frac{dr}{d\theta} = \theta^{-\frac{16}{9}}
   \]

   Finally, multiply both sides by \(r^{\frac{16}{9}}\):

   \[
   \frac{dr}{d\theta} = \left(\frac{r}{\theta}\right)^{-\frac{16}{9}}
   \]

Thus, the derivative \(\frac{dr}{d\theta}\) is:

\[
\boxed{\frac{dr}{d\theta} = \left(\frac{r}{\theta}\right)^{-\frac{16}{9}}}
\]

Would you like further details or any clarifications?

Here are some related questions for further exploration:
1. What is implicit differentiation, and how is it used in this problem?
2. How does the chain rule apply when differentiating functions of multiple variables?
3. Could you solve this equation for \(r\) explicitly before differentiating?
4. What are the steps to solve higher-order derivatives in this context?
5. How does the power rule help when differentiating negative exponents?

**Tip:** When differentiating implicit functions, always remember to apply the chain rule carefully when a dependent variable is involved.

Find dr/dθ for θ^(-7/9) - r^(-7/9) = 3.

Learn how to find the derivative dr/dθ for the equation θ^(-7/9) - r^(-7/9) = 3 using implicit differentiation and the chain rule. This step-by-step guide explores advanced calculus concepts involving negative exponents and differentiation rules, suitable for college-level students or advanced high school learners.

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