Differentiate `sin^(-1) [(3x+(4sqrt(1-x^(2))))/(5)] w.r.t.x`.

Answer

To differentiate the function \( y = \sin^{-1} \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right) \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function
We start with the given function:
\[ y = \sin^{-1} \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right) \]

### Step 2: Identify the inner function
Let:
\[ u = \frac{3x + 4\sqrt{1 - x^2}}{5} \]
Then, we can rewrite \( y \) as:
\[ y = \sin^{-1}(u) \]

### Step 3: Differentiate using the chain rule
Using the chain rule, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
The derivative of \( \sin^{-1}(u) \) is:
\[ \frac{dy}{du} = \frac{1}{\sqrt{1 - u^2}} \]

### Step 4: Differentiate \( u \)
Now we need to differentiate \( u \):
\[ u = \frac{3x + 4\sqrt{1 - x^2}}{5} \]
Differentiating \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{3}{5} + \frac{4}{5} \cdot \frac{d}{dx}(\sqrt{1 - x^2}) \]
Using the derivative of \( \sqrt{1 - x^2} \):
\[ \frac{d}{dx}(\sqrt{1 - x^2}) = \frac{-x}{\sqrt{1 - x^2}} \]
Thus:
\[ \frac{du}{dx} = \frac{3}{5} + \frac{4}{5} \cdot \left( \frac{-x}{\sqrt{1 - x^2}} \right) = \frac{3}{5} - \frac{4x}{5\sqrt{1 - x^2}} \]

### Step 5: Substitute back to find \(\frac{dy}{dx}\)
Now we substitute \( \frac{du}{dx} \) back into the expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot \left( \frac{3}{5} - \frac{4x}{5\sqrt{1 - x^2}} \right) \]

### Step 6: Calculate \( u^2 \)
Next, we need to calculate \( u^2 \):
\[ u^2 = \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right)^2 = \frac{(3x + 4\sqrt{1 - x^2})^2}{25} \]
Expanding this:
\[ u^2 = \frac{9x^2 + 24x\sqrt{1 - x^2} + 16(1 - x^2)}{25} = \frac{9x^2 + 16 - 16x^2 + 24x\sqrt{1 - x^2}}{25} = \frac{-7x^2 + 16 + 24x\sqrt{1 - x^2}}{25} \]

### Step 7: Substitute \( u^2 \) into \( \frac{dy}{dx} \)
Now substitute \( u^2 \) into the expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - \frac{-7x^2 + 16 + 24x\sqrt{1 - x^2}}{25}}} \cdot \left( \frac{3}{5} - \frac{4x}{5\sqrt{1 - x^2}} \right) \]

### Final Expression
This gives us the final expression for the derivative:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{\frac{25 - (-7x^2 + 16 + 24x\sqrt{1 - x^2})}{25}}} \cdot \left( \frac{3}{5} - \frac{4x}{5\sqrt{1 - x^2}} \right) \]