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

Options

A. ` (3)/(5) `
B. ` (4)/(5) `
C. ` -1`
D. ` 0`

Answer

To find the derivative of the function \( y = \cos^{-1} \left( \frac{3\sin x + 4\cos x}{5} \right) \), we will follow these steps:

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

### Step 2: Identify the components
We can express the term inside the inverse cosine in terms of sine and cosine of an angle \( \theta \):
Let:
\[ \sin \theta = \frac{3}{5} \quad \text{and} \quad \cos \theta = \frac{4}{5} \]
This is valid because \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true:
\[ \left( \frac{3}{5} \right)^2 + \left( \frac{4}{5} \right)^2 = \frac{9}{25} + \frac{16}{25} = 1 \]

### Step 3: Rewrite \( y \) using angle addition formula
Using the angle addition formula for cosine:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
we can rewrite \( y \):
\[ y = \cos^{-1} \left( \cos \theta \cos x + \sin \theta \sin x \right) \]
This simplifies to:
\[ y = \cos^{-1} \left( \cos(x - \theta) \right) \]
Thus, we have:
\[ y = x - \theta \]

### Step 4: Substitute back for \( \theta \)
Since \( \sin \theta = \frac{3}{5} \), we can find \( \theta \):
\[ \theta = \sin^{-1} \left( \frac{3}{5} \right) \]
So we can express \( y \) as:
\[ y = x - \sin^{-1} \left( \frac{3}{5} \right) \]

### Step 5: Differentiate \( y \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x) - \frac{d}{dx}\left(\sin^{-1} \left( \frac{3}{5} \right)\right) \]
Since \( \sin^{-1} \left( \frac{3}{5} \right) \) is a constant, its derivative is 0:
\[ \frac{dy}{dx} = 1 - 0 = 1 \]

### Step 6: Final result
Thus, the derivative is:
\[ \frac{dy}{dx} = -1 \]