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

Answer

To differentiate the function \( y = (\sin x)^{\cos^{-1} x} \) with respect to \( x \), we will use logarithmic differentiation. Here are the steps:

### Step 1: Take the natural logarithm of both sides
We start by taking the natural logarithm of both sides:
\[ \ln y = \ln((\sin x)^{\cos^{-1} x}) \]

### Step 2: Apply the logarithmic power rule
Using the property of logarithms, we can bring down the exponent:
\[ \ln y = \cos^{-1} x \cdot \ln(\sin x) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides. Using implicit differentiation on the left side and the product rule on the right side:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(\cos^{-1} x) \cdot \ln(\sin x) + \cos^{-1} x \cdot \frac{d}{dx}(\ln(\sin x)) \]

### Step 4: Differentiate \( \cos^{-1} x \)
The derivative of \( \cos^{-1} x \) is:
\[ \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}} \]

### Step 5: Differentiate \( \ln(\sin x) \)
The derivative of \( \ln(\sin x) \) is:
\[ \frac{d}{dx}(\ln(\sin x)) = \frac{1}{\sin x} \cdot \cos x = \frac{\cos x}{\sin x} = \cot x \]

### Step 6: Substitute the derivatives back into the equation
Substituting the derivatives into our equation gives:
\[ \frac{1}{y} \frac{dy}{dx} = -\frac{1}{\sqrt{1 - x^2}} \cdot \ln(\sin x) + \cos^{-1} x \cdot \cot x \]

### Step 7: Solve for \( \frac{dy}{dx} \)
Now, we multiply both sides by \( y \) to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \left( -\frac{1}{\sqrt{1 - x^2}} \cdot \ln(\sin x) + \cos^{-1} x \cdot \cot x \right) \]

### Step 8: Substitute back for \( y \)
Recall that \( y = (\sin x)^{\cos^{-1} x} \):
\[ \frac{dy}{dx} = (\sin x)^{\cos^{-1} x} \left( -\frac{1}{\sqrt{1 - x^2}} \cdot \ln(\sin x) + \cos^{-1} x \cdot \cot x \right) \]

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