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

Answer

To differentiate the function \( y = e^{\cos^{-1}(x^2)} \) with respect to \( x \), we will use the chain rule and the properties of derivatives. Here’s a step-by-step solution:

### Step 1: Identify the function
Let:
\[ y = e^{\cos^{-1}(x^2)} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( e^u \) with respect to \( x \) is \( e^u \cdot \frac{du}{dx} \), where \( u = \cos^{-1}(x^2) \).

Thus, we have:
\[ \frac{dy}{dx} = e^{\cos^{-1}(x^2)} \cdot \frac{d}{dx}(\cos^{-1}(x^2)) \]

### Step 3: Differentiate \( \cos^{-1}(x^2) \)
The derivative of \( \cos^{-1}(u) \) is given by:
\[ \frac{d}{du}(\cos^{-1}(u)) = -\frac{1}{\sqrt{1 - u^2}} \]
For \( u = x^2 \), we need to apply the chain rule again:
\[ \frac{d}{dx}(\cos^{-1}(x^2)) = -\frac{1}{\sqrt{1 - (x^2)^2}} \cdot \frac{d}{dx}(x^2) \]
Calculating \( \frac{d}{dx}(x^2) \):
\[ \frac{d}{dx}(x^2) = 2x \]
Thus, we have:
\[ \frac{d}{dx}(\cos^{-1}(x^2)) = -\frac{1}{\sqrt{1 - x^4}} \cdot 2x = -\frac{2x}{\sqrt{1 - x^4}} \]

### Step 4: Substitute back into the derivative
Now substituting back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = e^{\cos^{-1}(x^2)} \cdot \left(-\frac{2x}{\sqrt{1 - x^4}}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = -\frac{2x e^{\cos^{-1}(x^2)}}{\sqrt{1 - x^4}} \]

### Final Answer
Thus, the derivative of \( y = e^{\cos^{-1}(x^2)} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{2x e^{\cos^{-1}(x^2)}}{\sqrt{1 - x^4}} \]