To solve the given problem, we will find the derivative $\frac{dy}{dt}$ for $y = (1 + \sin 10t)^{-4}$.

Step-by-step solution:

1. Use the chain rule: $\frac{dy}{dt} = \frac{d}{dt}\left[(1 + \sin 10t)^{-4}\right]$ First, apply the power rule: $\frac{dy}{dt} = -4(1 + \sin 10t)^{-5} \cdot \frac{d}{dt}(1 + \sin 10t)$

2. Differentiate $(1 + \sin 10t)$: $\frac{d}{dt}(1 + \sin 10t) = 0 + \cos 10t \cdot \frac{d}{dt}(10t) = 10 \cos 10t$

3. Substitute back into the derivative: $\frac{dy}{dt} = -4(1 + \sin 10t)^{-5} \cdot 10 \cos 10t$

4. Simplify: $\frac{dy}{dt} = -40 \cos 10t (1 + \sin 10t)^{-5}$

Correct Answer:

The correct choice is A: $-40 \cos 10t (1 + \sin 10t)^{-5}$.

Let me know if you need further explanation or additional steps! Here are some related questions to deepen your understanding:

Related Questions:

1. How does the chain rule work with nested functions like this example?
2. What is the general formula for differentiating $(f(t))^n$?
3. How does the derivative of sine or cosine change with a coefficient like $10t$?
4. Could you rewrite the result using a different notation (e.g., $y'$ or Leibniz's notation)?
5. How would the solution differ if the power were positive (e.g., $+4$) instead of negative?

Tip:

When differentiating composite functions, always identify the outer and inner functions clearly to apply the chain rule correctly.