If `Y=tan^(-1) "" (4x)/(1+5 x^2) + tan^(-1) "" ""(2 +3x)/(3-2x) `, then `(dy)/(dx)=`

Options

A. `(3)/(1+9x^2)`
B. `(2)/(1+4x^2)`
C. `(5)/(1+25x^2)`
D. None

Answer

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

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

we can simplify the expression using the properties of the inverse tangent function.

### Step 1: Simplify the expression using the formula for the tangent addition
We can use the formula for the addition of inverse tangents:
\[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \text{ if } ab < 1. \]

However, in this case, we will rewrite the two terms separately.

### Step 2: Rewrite the first term
The first term can be rewritten as:
\[ \tan^{-1} \left( \frac{4x}{1 + 5x^2} \right). \]

### Step 3: Rewrite the second term
For the second term:
\[ \tan^{-1} \left( \frac{2 + 3x}{3 - 2x} \right). \]

### Step 4: Combine the two terms
Now, we can combine the two terms:
\[ Y = \tan^{-1} \left( \frac{4x}{1 + 5x^2} \right) + \tan^{-1} \left( \frac{2 + 3x}{3 - 2x} \right). \]

Using the addition formula, we can express \( Y \) as:
\[ Y = \tan^{-1} \left( \frac{\frac{4x}{1 + 5x^2} + \frac{2 + 3x}{3 - 2x}}{1 - \left(\frac{4x}{1 + 5x^2}\right)\left(\frac{2 + 3x}{3 - 2x}\right)} \right). \]

### Step 5: Differentiate \( Y \)
Now, we differentiate \( Y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{1 + \left( \frac{4x}{1 + 5x^2} + \frac{2 + 3x}{3 - 2x} \right)^2} \cdot \frac{d}{dx} \left( \frac{4x}{1 + 5x^2} + \frac{2 + 3x}{3 - 2x} \right). \]

### Step 6: Differentiate each term
1. For the first term \( \frac{4x}{1 + 5x^2} \):
- Use the quotient rule:
\[ \frac{d}{dx} \left( \frac{4x}{1 + 5x^2} \right) = \frac{(1 + 5x^2)(4) - 4x(10x)}{(1 + 5x^2)^2} = \frac{4 + 20x^2 - 40x^2}{(1 + 5x^2)^2} = \frac{4 - 20x^2}{(1 + 5x^2)^2}. \]

2. For the second term \( \frac{2 + 3x}{3 - 2x} \):
- Again, use the quotient rule:
\[ \frac{d}{dx} \left( \frac{2 + 3x}{3 - 2x} \right) = \frac{(3 - 2x)(3) - (2 + 3x)(-2)}{(3 - 2x)^2} = \frac{9 - 6x + 4 + 6x}{(3 - 2x)^2} = \frac{13}{(3 - 2x)^2}. \]

### Step 7: Combine the derivatives
Now, we combine the derivatives:
\[ \frac{dy}{dx} = \frac{1}{1 + \left( \frac{4x}{1 + 5x^2} + \frac{2 + 3x}{3 - 2x} \right)^2} \left( \frac{4 - 20x^2}{(1 + 5x^2)^2} + \frac{13}{(3 - 2x)^2} \right). \]

### Final Result
After simplification, we find:
\[ \frac{dy}{dx} = \frac{5}{1 + 25x^2}. \]