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To solve the problem \( \frac{dy}{dx} \) for \( y = (\sin x)^{-3} \), we'll use the chain rule and some differentiation techniques.

Given:
\[
y = (\sin x)^{-3}
\]

### Step 1: Rewrite the function for easier differentiation.
\[
y = \sin^{-3} x
\]

### Step 2: Apply the chain rule.
The chain rule states:
\[
\frac{dy}{dx} = \frac{d}{dx} \left[\sin^{-3} x\right] = -3 \cdot \sin^{-4} x \cdot \frac{d}{dx}[\sin x]
\]

### Step 3: Differentiate \(\sin x\) with respect to \(x\).
\[
\frac{d}{dx}[\sin x] = \cos x
\]

### Step 4: Substitute the derivative of \(\sin x\) back into the equation.
\[
\frac{dy}{dx} = -3 \cdot \sin^{-4} x \cdot \cos x
\]

### Step 5: Simplify the expression.
\[
\frac{dy}{dx} = -3 \cdot \frac{\cos x}{\sin^4 x}
\]

So the derivative is:
\[
\boxed{\frac{dy}{dx} = -\frac{3\cos x}{\sin^4 x}}
\]

Would you like more details on any step or have any questions?

Here are some related questions to expand your understanding:

1. How does the chain rule apply to more complex functions?
2. What is the derivative of \(\sin^3 x\) using the chain rule?
3. Can you explain the product rule and when it would be applicable?
4. How would you find the second derivative of \(y = (\sin x)^{-3}\)?
5. What happens if we differentiate \(y = (\cos x)^{-3}\) instead?

**Tip:** When differentiating functions with negative exponents, rewriting them can simplify the application of differentiation rules.

Learn how to find the derivative of y = (sinx)^{-3} with a step-by-step solution using the chain rule. This problem covers key differentiation concepts and is suitable for students in Grades 11-12.

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