If ` y =cot ^(-1) ((3+4tan x )/( 4-3tan x) ) ,then (dy)/(dx)=`

Options

A. ` -3`
B. 4
C. `-1`
D. 1

Answer

To solve the problem \( y = \cot^{-1} \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right) \) and find \( \frac{dy}{dx} \), we can follow these steps:

### Step 1: Rewrite \( y \) using the identity for cotangent
We know that \( \cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x) \). Therefore, we can rewrite \( y \) as:
\[ y = \frac{\pi}{2} - \tan^{-1} \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right) \]

### Step 2: Simplify the argument of \( \tan^{-1} \)
Next, we simplify the argument of the \( \tan^{-1} \) function:
\[ y = \frac{\pi}{2} - \tan^{-1} \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right) \]

### Step 3: Use the identity for \( \tan^{-1} \)
Using the identity for the sum of two \( \tan^{-1} \) functions:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \]
we can see that we can express \( \tan^{-1} \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right) \) in a different form, but in this case, we will proceed directly to differentiation.

### Step 4: Differentiate \( y \)
Differentiating \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 0 - \frac{1}{1 + \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right)^2} \cdot \frac{d}{dx} \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right) \]

### Step 5: Differentiate the fraction
Using the quotient rule:
\[ \frac{d}{dx} \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right) = \frac{(4)(4 - 3\tan x) - (3 + 4\tan x)(-3)}{(4 - 3\tan x)^2} \]

### Step 6: Simplify the derivative
Calculating the numerator:
\[ = 16 - 12\tan x + 9 + 12\tan x = 25 \]
Thus:
\[ \frac{d}{dx} \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right) = \frac{25}{(4 - 3\tan x)^2} \]

### Step 7: Substitute back into the derivative
Now substituting back into the derivative of \( y \):
\[ \frac{dy}{dx} = -\frac{1}{1 + \left( \frac{3 + 4\tan x}{4 - 3\tan x} \right)^2} \cdot \frac{25}{(4 - 3\tan x)^2} \]

### Step 8: Final simplification
The final expression will yield:
\[ \frac{dy}{dx} = -\frac{25}{(4 - 3\tan x)^2 + (3 + 4\tan x)^2} \]
After simplifying, we find that:
\[ \frac{dy}{dx} = -1 \]

### Final Answer
\[ \frac{dy}{dx} = -1 \]