`sin^(-1)3x`

Answer

To find the derivative of the function \( y = \sin^{-1}(3x) \) with respect to \( x \), we will follow these steps:

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

### Step 2: Use the derivative formula for \( \sin^{-1}(x) \)
The derivative of \( \sin^{-1}(x) \) with respect to \( x \) is given by:
\[ \frac{d}{dx} \sin^{-1}(x) = \frac{1}{\sqrt{1 - x^2}} \]

### Step 3: Apply the chain rule
Since we have \( \sin^{-1}(3x) \), we need to apply the chain rule. The chain rule states that if \( y = f(g(x)) \), then:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
Here, \( f(x) = \sin^{-1}(x) \) and \( g(x) = 3x \).

### Step 4: Differentiate \( g(x) \)
First, we find the derivative of \( g(x) = 3x \):
\[ g'(x) = 3 \]

### Step 5: Substitute into the derivative formula
Now, we substitute \( g(x) = 3x \) into the derivative formula for \( \sin^{-1}(x) \):
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (3x)^2}} \cdot g'(x) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - 9x^2}} \cdot 3 \]

### Step 6: Final expression for the derivative
Thus, we have:
\[ \frac{dy}{dx} = \frac{3}{\sqrt{1 - 9x^2}} \]

### Final Answer
The derivative of \( y = \sin^{-1}(3x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{3}{\sqrt{1 - 9x^2}} \]

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