Find `dy/dx if siny=x^3`

Answer

To find \(\frac{dy}{dx}\) for the equation \(\sin y = x^3\), we will use implicit differentiation. Here are the steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sin y = x^3 \]

2. **Differentiate both sides with respect to \(x\)**:
- The left side involves \(y\), which is a function of \(x\), so we use the chain rule:
\[ \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx} \]
- The right side is a simple power function:
\[ \frac{d}{dx}(x^3) = 3x^2 \]

3. **Set the derivatives equal to each other**:
\[ \cos y \cdot \frac{dy}{dx} = 3x^2 \]

4. **Solve for \(\frac{dy}{dx}\)**:
- To isolate \(\frac{dy}{dx}\), divide both sides by \(\cos y\):
\[ \frac{dy}{dx} = \frac{3x^2}{\cos y} \]

5. **Use the identity for secant**:
- Recall that \(\sec y = \frac{1}{\cos y}\), so we can rewrite the equation:
\[ \frac{dy}{dx} = 3x^2 \sec y \]

### Final Answer:
\[ \frac{dy}{dx} = 3x^2 \sec y \]