`sin^(-1)3x`

Answer

To differentiate \( y = \sin^{-1}(3x) \) with respect to \( x \), we will use the chain rule and the formula for the derivative of the inverse sine function.

### Step-by-Step Solution:

1. **Identify the function**:
Let \( y = \sin^{-1}(3x) \).

2. **Use the derivative formula**:
The derivative of \( \sin^{-1}(u) \) with respect to \( u \) is given by:
\[ \frac{dy}{du} = \frac{1}{\sqrt{1 - u^2}} \]
where \( u = 3x \).

3. **Differentiate \( u \)**:
Now, we need to find \( \frac{du}{dx} \):
\[ u = 3x \implies \frac{du}{dx} = 3 \]

4. **Apply the chain rule**:
According to the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

5. **Substitute the derivatives**:
Substitute \( u = 3x \) into the derivative formula:
\[ \frac{dy}{du} = \frac{1}{\sqrt{1 - (3x)^2}} = \frac{1}{\sqrt{1 - 9x^2}} \]
Now, substituting back into the chain rule:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - 9x^2}} \cdot 3 \]

6. **Final expression**:
Thus, the derivative of \( y = \sin^{-1}(3x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{3}{\sqrt{1 - 9x^2}} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{3}{\sqrt{1 - 9x^2}} \]

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