Differentiate the following w.r.t. x :
`sin(2sin^(-1)x)`

Answer

To differentiate the function \( y = \sin(2 \sin^{-1} x) \) with respect to \( x \), we can use the chain rule. Here’s a step-by-step solution:

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

### Step 2: Differentiate \( y \) with respect to \( u \)
Using the derivative of sine, we have:
\[ \frac{dy}{du} = \cos(u) \]

### Step 3: Differentiate \( u \) with respect to \( x \)
Now we need to find \( \frac{du}{dx} \). The derivative of \( \sin^{-1} x \) is:
\[ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} \]
Thus, differentiating \( u \):
\[ u = 2 \sin^{-1} x \implies \frac{du}{dx} = 2 \cdot \frac{1}{\sqrt{1 - x^2}} = \frac{2}{\sqrt{1 - x^2}} \]

### Step 4: Apply the chain rule
Now, we can use the chain rule to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(u) \cdot \frac{du}{dx} \]
Substituting \( u = 2 \sin^{-1} x \):
\[ \frac{dy}{dx} = \cos(2 \sin^{-1} x) \cdot \frac{2}{\sqrt{1 - x^2}} \]

### Step 5: Simplify \( \cos(2 \sin^{-1} x) \)
Using the double angle identity for cosine:
\[ \cos(2\theta) = 1 - 2\sin^2(\theta) \]
Let \( \theta = \sin^{-1} x \), then \( \sin(\theta) = x \). Therefore:
\[ \cos(2 \sin^{-1} x) = 1 - 2x^2 \]

### Step 6: Substitute back into the derivative
Now substituting back, we have:
\[ \frac{dy}{dx} = (1 - 2x^2) \cdot \frac{2}{\sqrt{1 - x^2}} \]

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

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