derivative of `sin^(-1)(ax)`

Answer

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

### Step 1: Identify the function and its derivative formula
The given function is \( y = \sin^{-1}(ax) \). We know that the derivative of \( \sin^{-1}(x) \) is given by:
\[ \frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1 - x^2}} \]

### Step 2: Apply the chain rule
Since \( ax \) is a function of \( x \), we will use the chain rule to differentiate \( \sin^{-1}(ax) \). According to the chain rule:
\[ \frac{d}{dx}(\sin^{-1}(u)) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = ax \).

### Step 3: Differentiate the inner function
Now, we need to find \( \frac{du}{dx} \) where \( u = ax \). The derivative of \( u = ax \) with respect to \( x \) is:
\[ \frac{du}{dx} = a \]
because \( a \) is a constant.

### Step 4: Substitute back into the derivative formula
Now we substitute \( u = ax \) and \( \frac{du}{dx} = a \) into the derivative formula:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (ax)^2}} \cdot a \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = \frac{a}{\sqrt{1 - a^2x^2}} \]

### Final Answer
Thus, the derivative of \( \sin^{-1}(ax) \) is:
\[ \frac{d}{dx}(\sin^{-1}(ax)) = \frac{a}{\sqrt{1 - a^2x^2}} \]

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