If `y =tan ^(-1)((x ^(1//3) -a^(1//3))/(1+x ^(1//3)a ^(1//3))), x gt 0, a gt 0, then (dy)/(dx)` is:

Options

A. `(1)/(x ^(2//3)(1+ x ^(2//3)))`
B. `(3)/(x ^(2//3)(1+ x ^(2//3)))`
C. `(1)/(3x ^(2//3)(1+ x ^(2//3)))`
D. `(1)/(3x ^(1//3)(1+ x ^(2//3)))`

Answer

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

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

where \( x > 0 \) and \( a > 0 \), we can follow these steps:

### Step 1: Rewrite the function using the formula for the difference of arctangents

We can use the identity:

\[ \tan^{-1}(u) - \tan^{-1}(v) = \tan^{-1}\left(\frac{u - v}{1 + uv}\right) \]

In our case, let \( u = x^{1/3} \) and \( v = a^{1/3} \). Thus, we can write:

\[ y = \tan^{-1}(x^{1/3}) - \tan^{-1}(a^{1/3}) \]

### Step 2: Differentiate \( y \) with respect to \( x \)

Now, we differentiate \( y \):

\[ \frac{dy}{dx} = \frac{d}{dx} \left(\tan^{-1}(x^{1/3})\right) - \frac{d}{dx} \left(\tan^{-1}(a^{1/3})\right) \]

Since \( a^{1/3} \) is a constant, its derivative is 0. Therefore:

\[ \frac{dy}{dx} = \frac{d}{dx} \left(\tan^{-1}(x^{1/3})\right) \]

### Step 3: Use the derivative of the arctangent function

The derivative of \( \tan^{-1}(u) \) is given by:

\[ \frac{d}{dx} \tan^{-1}(u) = \frac{1}{1 + u^2} \cdot \frac{du}{dx} \]

Here, \( u = x^{1/3} \). We need to find \( \frac{du}{dx} \):

\[ u = x^{1/3} \quad \Rightarrow \quad \frac{du}{dx} = \frac{1}{3} x^{-2/3} \]

### Step 4: Substitute back into the derivative formula

Now we substitute \( u \) and \( \frac{du}{dx} \) back into the derivative formula:

\[ \frac{dy}{dx} = \frac{1}{1 + (x^{1/3})^2} \cdot \frac{1}{3} x^{-2/3} \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{1}{1 + x^{2/3}} \cdot \frac{1}{3} x^{-2/3} \]

### Step 5: Final expression

Thus, we can write:

\[ \frac{dy}{dx} = \frac{1}{3} \cdot \frac{x^{-2/3}}{1 + x^{2/3}} = \frac{1}{3} \cdot \frac{1}{x^{2/3}(1 + x^{2/3})} \]

### Final Answer

The final expression for \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{1}{3} \cdot \frac{1}{x^{2/3}(1 + x^{2/3})} \]