If ` y = tan^(-1)((sin 3x - cos 3x )/(sin 3x + cos 3x))" then" (dy)/(dx) = `

Options

A. 0
B. 2
C. 1
D. 3

Explanation

To solve the problem, we need to differentiate the function given by:

\[ y = \tan^{-1}\left(\frac{\sin 3x - \cos 3x}{\sin 3x + \cos 3x}\right) \]

### Step 1: Simplify the Argument of the Inverse Tangent

First, we can simplify the expression inside the inverse tangent function. We divide both the numerator and the denominator by \(\cos 3x\):

\[ y = \tan^{-1}\left(\frac{\frac{\sin 3x}{\cos 3x} - 1}{\frac{\sin 3x}{\cos 3x} + 1}\right) \]

This simplifies to:

\[ y = \tan^{-1\left(\frac{\tan 3x - 1}{\tan 3x + 1}\right) \]

### Step 2: Use the Tangent Subtraction Formula

The expression \(\frac{\tan A - \tan B}{1 + \tan A \tan B} = \tan(A - B)\) can be used here. Let \(A = 3x\) and \(B = \frac{\pi}{4}\) (since \(\tan \frac{\pi}{4} = 1\)):

\[ y = \tan^{-1}\left(\tan\left(3x - \frac{\pi}{4}\right)\right) \]

### Step 3: Simplify Further

Since the tangent and inverse tangent functions are inverses, we can simplify:

\[ y = 3x - \frac{\pi}{4} \]

### Step 4: Differentiate with Respect to \(x\)

Now, we differentiate \(y\) with respect to \(x\):

\[ \frac{dy}{dx} = \frac{d}{dx}(3x - \frac{\pi}{4}) = 3 \]

### Final Answer

Thus, the derivative \(\frac{dy}{dx}\) is:

\[ \frac{dy}{dx} = 3 \]

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