Differentiate `sin^(-1)((2^(x+1))/(1+4^(x)))` w.r.t. x.

Answer

To differentiate the function \( y = \sin^{-1}\left(\frac{2^{x+1}}{1 + 4^x}\right) \) with respect to \( x \), we will follow these steps:

### Step 1: Rewrite the Function
First, we can rewrite the function in a more manageable form:
\[ y = \sin^{-1}\left(\frac{2^{x+1}}{1 + 4^x}\right) = \sin^{-1}\left(\frac{2 \cdot 2^x}{1 + (2^2)^x}\right) = \sin^{-1}\left(\frac{2 \cdot 2^x}{1 + 2^{2x}}\right) \]

### Step 2: Use the Chain Rule
To differentiate \( y \), we will apply the chain rule. The derivative of \( \sin^{-1}(u) \) is given by:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]
where \( u = \frac{2 \cdot 2^x}{1 + 2^{2x}} \).

### Step 3: Differentiate \( u \)
Next, we need to find \( \frac{du}{dx} \). We can use the quotient rule for differentiation:
\[ u = \frac{2 \cdot 2^x}{1 + 2^{2x}} \]
Let \( a = 2 \cdot 2^x \) and \( b = 1 + 2^{2x} \). The quotient rule states:
\[ \frac{du}{dx} = \frac{b \frac{da}{dx} - a \frac{db}{dx}}{b^2} \]

Calculating \( \frac{da}{dx} \) and \( \frac{db}{dx} \):
- \( a = 2 \cdot 2^x \) implies \( \frac{da}{dx} = 2^x \ln(2) \)
- \( b = 1 + 2^{2x} \) implies \( \frac{db}{dx} = 2^{2x} \cdot 2 \ln(2) = 2^{2x + 1} \ln(2) \)

Now substituting back into the quotient rule:
\[ \frac{du}{dx} = \frac{(1 + 2^{2x}) \cdot (2^x \ln(2)) - (2 \cdot 2^x) \cdot (2^{2x + 1} \ln(2))}{(1 + 2^{2x})^2} \]

### Step 4: Simplify \( \frac{du}{dx} \)
This simplifies to:
\[ \frac{du}{dx} = \frac{2^x \ln(2) \left(1 + 2^{2x} - 2 \cdot 2^{3x}\right)}{(1 + 2^{2x})^2} \]

### Step 5: Substitute \( u \) and \( \frac{du}{dx} \) into the Chain Rule
Now we can substitute \( u \) and \( \frac{du}{dx} \) back into the derivative of \( y \):
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - \left(\frac{2 \cdot 2^x}{1 + 2^{2x}}\right)^2}} \cdot \frac{du}{dx} \]

### Step 6: Final Expression
Thus, the final expression for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - \left(\frac{2 \cdot 2^x}{1 + 2^{2x}}\right)^2}} \cdot \frac{2^x \ln(2) \left(1 + 2^{2x} - 2 \cdot 2^{3x}\right)}{(1 + 2^{2x})^2} \]