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 $t$.