What is the derivative of `cos^(-1)((2cosx+3sinx)/(sqrt(13)))` ?

Options

A. `(1)/(sqrt(1-x^(2)))`
B. `-(1)/(sqrt(1-x^(2)))`
C. 0
D. 1

Answer

To find the derivative of \( y = \cos^{-1}\left(\frac{2\cos x + 3\sin x}{\sqrt{13}}\right) \), we can follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
We start with the expression:
\[ y = \cos^{-1}\left(\frac{2\cos x + 3\sin x}{\sqrt{13}}\right) \]
We can recognize that this expression can be rewritten using the cosine of a sum formula.

### Step 2: Identify the angle
Let \( \alpha \) be an angle such that:
\[ \cos \alpha = \frac{2}{\sqrt{13}} \quad \text{and} \quad \sin \alpha = \frac{3}{\sqrt{13}} \]
This means that we can express \( \frac{2\cos x + 3\sin x}{\sqrt{13}} \) as:
\[ \cos(\alpha - x) \]
Thus, we can rewrite \( y \) as:
\[ y = \cos^{-1}(\cos(\alpha - x)) \]

### Step 3: Simplify the expression
Using the property of the inverse cosine function, we have:
\[ y = \alpha - x \]

### Step 4: Differentiate with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d(\alpha - x)}{dx} = \frac{d\alpha}{dx} - \frac{dx}{dx} \]
Since \( \alpha \) is a constant (it does not depend on \( x \)), its derivative is 0:
\[ \frac{dy}{dx} = 0 - 1 = -1 \]

### Conclusion
Thus, the derivative of \( y \) is:
\[ \frac{dy}{dx} = -1 \]