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To solve the given problem, we need to find \(\frac{dx}{dt}\) at \(x = 5\) given the equations \(y = -3x^2 - 5\) and \(\frac{dy}{dt} = 3\).

### Steps to solve:

1. **Differentiate the given equation \(y = -3x^2 - 5\) with respect to \(t\):**
   \[
   \frac{dy}{dt} = \frac{d}{dt}(-3x^2 - 5)
   \]
   Using the chain rule, we get:
   \[
   \frac{dy}{dt} = -6x \frac{dx}{dt}
   \]

2. **Substitute the given values:**
   - \(x = 5\)
   - \(\frac{dy}{dt} = 3\)

   So, the equation becomes:
   \[
   3 = -6(5) \frac{dx}{dt}
   \]

3. **Solve for \(\frac{dx}{dt}\):**
   \[
   3 = -30 \frac{dx}{dt}
   \]
   \[
   \frac{dx}{dt} = \frac{3}{-30} = -\frac{1}{10}
   \]

Thus, the solution is:
\[
\boxed{-\frac{1}{10}}
\]

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. How do you apply the chain rule in calculus?
2. What is implicit differentiation, and when is it used?
3. How do you find the rate of change of a function with respect to another variable?
4. Can you provide an example of a related rates problem?
5. What are some real-life applications of related rates problems?
6. How do you interpret the result of a derivative in a physical context?
7. What is the difference between partial derivatives and ordinary derivatives?
8. How does the derivative of a composite function work?

**Tip:** When differentiating implicitly, remember to apply the chain rule correctly and treat each variable as a function of \(t\).

Learn how to calculate the rate of change using the chain rule in calculus. This problem involves finding \(\frac{dx}{dt}\) at \(x = 5\) given the equation \(y = -3x^2 - 5\) and \(\frac{dy}{dt} = 3\). Detailed step-by-step solution provided.

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