Differentiate `sin^(-1)(4xsqrt(1-4x^(2)))w.r.t.sqrt(1-4x^(2))`, if `x in(-(1)/(2sqrt2),(1)/(2sqrt2))`

Answer

To differentiate \( \sin^{-1}(4x \sqrt{1 - 4x^2}) \) with respect to \( \sqrt{1 - 4x^2} \), we will follow these steps:

### Step 1: Define the function
Let:
\[ y = \sin^{-1}(4x \sqrt{1 - 4x^2}) \]
We need to differentiate \( y \) with respect to \( z = \sqrt{1 - 4x^2} \).

### Step 2: Use the chain rule
Using the chain rule, we can express the derivative as:
\[ \frac{dy}{dz} = \frac{dy}{dx} \cdot \frac{dx}{dz} \]

### Step 3: Differentiate \( y \) with respect to \( x \)
To find \( \frac{dy}{dx} \), we use the derivative of the inverse sine function:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (4x \sqrt{1 - 4x^2})^2}} \cdot \frac{d}{dx}(4x \sqrt{1 - 4x^2}) \]

Now, we need to differentiate \( 4x \sqrt{1 - 4x^2} \):
Using the product rule:
\[ \frac{d}{dx}(4x \sqrt{1 - 4x^2}) = 4 \sqrt{1 - 4x^2} + 4x \cdot \frac{1}{2\sqrt{1 - 4x^2}} \cdot (-8x) \]
Simplifying:
\[ = 4 \sqrt{1 - 4x^2} - \frac{16x^2}{2\sqrt{1 - 4x^2}} = 4 \sqrt{1 - 4x^2} - \frac{8x^2}{\sqrt{1 - 4x^2}} \]
Combining the terms:
\[ = \frac{4(1 - 4x^2) - 8x^2}{\sqrt{1 - 4x^2}} = \frac{4 - 32x^2}{\sqrt{1 - 4x^2}} \]

Thus:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (4x \sqrt{1 - 4x^2})^2}} \cdot \frac{4 - 32x^2}{\sqrt{1 - 4x^2}} \]

### Step 4: Differentiate \( z \) with respect to \( x \)
Now, we differentiate \( z = \sqrt{1 - 4x^2} \):
\[ \frac{dz}{dx} = \frac{1}{2\sqrt{1 - 4x^2}} \cdot (-8x) = -\frac{4x}{\sqrt{1 - 4x^2}} \]

### Step 5: Combine the derivatives
Now we can combine these results to find \( \frac{dy}{dz} \):
\[ \frac{dy}{dz} = \frac{dy}{dx} \cdot \frac{dx}{dz} = \frac{dy}{dx} \cdot \frac{1}{\frac{dz}{dx}} = \frac{dy}{dx} \cdot \left(-\frac{\sqrt{1 - 4x^2}}{4x}\right) \]

Substituting \( \frac{dy}{dx} \):
\[ \frac{dy}{dz} = \left(\frac{1}{\sqrt{1 - (4x \sqrt{1 - 4x^2})^2}} \cdot \frac{4 - 32x^2}{\sqrt{1 - 4x^2}}\right) \cdot \left(-\frac{\sqrt{1 - 4x^2}}{4x}\right) \]

### Step 6: Simplify the expression
This simplifies to:
\[ \frac{dy}{dz} = -\frac{(4 - 32x^2)}{4x \sqrt{1 - (4x \sqrt{1 - 4x^2})^2}} \]

### Final Result
Thus, the derivative of \( \sin^{-1}(4x \sqrt{1 - 4x^2}) \) with respect to \( \sqrt{1 - 4x^2} \) is:
\[ \frac{dy}{dz} = -\frac{(4 - 32x^2)}{4x \sqrt{1 - (4x \sqrt{1 - 4x^2})^2}} \]