Find the derivative of the following function with respect to 'x'
`sin^-1(1-2x^2)`

Answer

To find the derivative of the function \( y = \sin^{-1}(1 - 2x^2) \) with respect to \( x \), we will follow these steps:

### Step 1: Set up the function
Let:
\[ y = \sin^{-1}(1 - 2x^2) \]
We will use the chain rule to differentiate this function.

### Step 2: Identify the inner function
Let:
\[ u = 1 - 2x^2 \]
Then we can rewrite \( y \) as:
\[ y = \sin^{-1}(u) \]

### Step 3: Differentiate using the chain rule
The derivative of \( \sin^{-1}(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \frac{1}{\sqrt{1 - u^2}} \]
Now we need to find \( \frac{du}{dx} \).

### Step 4: Differentiate the inner function
Differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(1 - 2x^2) = -4x \]

### Step 5: Apply the chain rule
Now, using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot (-4x) \]

### Step 6: Substitute back for \( u \)
Now, substitute \( u = 1 - 2x^2 \) back into the equation:
\[ \frac{dy}{dx} = \frac{-4x}{\sqrt{1 - (1 - 2x^2)^2}} \]

### Step 7: Simplify the expression
Now simplify the expression under the square root:
\[ 1 - (1 - 2x^2)^2 = 1 - (1 - 4x^2 + 4x^4) = 4x^2 - 4x^4 \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{-4x}{\sqrt{4x^2 - 4x^4}} = \frac{-4x}{\sqrt{4x^2(1 - x^2)}} = \frac{-4x}{2\sqrt{x^2(1 - x^2)}} = \frac{-2}{\sqrt{1 - x^2}} \]

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

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