If ` y=sin ^(-1) ((3x +4 sqrt(1-x^(2)))/( 5)),then (dy)/(dx)=`

Options

A. ` (-1)/(sqrt( 1-x^(2)))`
B. ` (1)/(sqrt( 1-x^(2)))`
C. ` (-2)/(sqrt( 1-x^(2)))`
D. ` (2)/(sqrt( 1-x^(2)))`

Answer

To find the derivative \( \frac{dy}{dx} \) for the function

\[ y = \sin^{-1} \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right), \]

we will follow these steps:

### Step 1: Set up the equation
We start with the equation given:

\[ y = \sin^{-1} \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right). \]

### Step 2: Differentiate using the chain rule
Using the chain rule, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right)^2}} \cdot \frac{d}{dx} \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right). \]

### Step 3: Differentiate the inner function
Now we need to differentiate the inner function \( \frac{3x + 4\sqrt{1 - x^2}}{5} \):

\[ \frac{d}{dx} \left( 3x + 4\sqrt{1 - x^2} \right) = 3 + 4 \cdot \frac{1}{2\sqrt{1 - x^2}} \cdot (-2x) = 3 - \frac{4x}{\sqrt{1 - x^2}}. \]

Thus,

\[ \frac{d}{dx} \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right) = \frac{1}{5} \left( 3 - \frac{4x}{\sqrt{1 - x^2}} \right). \]

### Step 4: Substitute back into the derivative
Now we substitute back into the derivative expression:

\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right)^2}} \cdot \frac{1}{5} \left( 3 - \frac{4x}{\sqrt{1 - x^2}} \right). \]

### Step 5: Simplify the expression
Next, we simplify the term \( 1 - \left( \frac{3x + 4\sqrt{1 - x^2}}{5} \right)^2 \):

Let \( u = \frac{3x + 4\sqrt{1 - x^2}}{5} \).

Then,

\[ u^2 = \frac{(3x + 4\sqrt{1 - x^2})^2}{25} = \frac{9x^2 + 24x\sqrt{1 - x^2} + 16(1 - x^2)}{25}. \]

Now, we can find \( 1 - u^2 \):

\[ 1 - u^2 = 1 - \frac{9x^2 + 24x\sqrt{1 - x^2} + 16(1 - x^2)}{25} = \frac{25 - (9x^2 + 24x\sqrt{1 - x^2} + 16 - 16x^2)}{25} = \frac{9 - 24x\sqrt{1 - x^2}}{25}. \]

### Final expression for \( \frac{dy}{dx} \)
Putting it all together, we have:

\[ \frac{dy}{dx} = \frac{1}{\sqrt{\frac{9 - 24x\sqrt{1 - x^2}}{25}}} \cdot \frac{1}{5} \left( 3 - \frac{4x}{\sqrt{1 - x^2}} \right). \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{5}{\sqrt{9 - 24x\sqrt{1 - x^2}}} \cdot \frac{1}{5} \left( 3 - \frac{4x}{\sqrt{1 - x^2}} \right) = \frac{3 - \frac{4x}{\sqrt{1 - x^2}}}{\sqrt{9 - 24x\sqrt{1 - x^2}}}. \]