`(d)/(dx)[sin^(-1){cos(x^(2)-2)}]=`

Options

A. x
B. `-x`
C. 2x
D. `-2x`

Answer

To solve the problem \((d)/(dx)[\sin^{-1}(\cos(x^2 - 2))]\), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression \(\sin^{-1}(\cos(x^2 - 2))\). We can use the identity that relates cosine and sine:
\[ \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \]
Thus, we can rewrite \(\sin^{-1}(\cos(x^2 - 2))\) as:
\[ \sin^{-1}(\cos(x^2 - 2)) = \sin^{-1}\left(\sin\left(\frac{\pi}{2} - (x^2 - 2)\right)\right) \]
This simplifies to:
\[ \frac{\pi}{2} - (x^2 - 2) = \frac{\pi}{2} - x^2 + 2 \]

### Step 2: Differentiate the Expression
Now we differentiate the expression \(\frac{\pi}{2} - x^2 + 2\):
\[ \frac{d}{dx}\left(\frac{\pi}{2} - x^2 + 2\right) \]
The derivative of a constant is zero, so we only need to differentiate \(-x^2\):
\[ \frac{d}{dx}(-x^2) = -2x \]

### Step 3: Combine the Results
Thus, the derivative of \(\sin^{-1}(\cos(x^2 - 2))\) is:
\[ -2x \]

### Final Answer
Therefore, the final result is:
\[ \frac{d}{dx}[\sin^{-1}(\cos(x^2 - 2))] = -2x \]

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