if `x=(1+t)/t^3 ,y=3/(2t^2)+2/t` satisfies `f(x)*{(dy)/(dx)}^3=1+(dy)/(dx)` then `f(x)` is:

Options

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

Explanation

To solve the problem, we need to find the function \( f(x) \) given the relationship:

\[ f(x) \left( \frac{dy}{dx} \right)^3 = 1 + \frac{dy}{dx} \]

where \( x = \frac{1+t}{t^3} \) and \( y = \frac{3}{2t^2} + \frac{2}{t} \).

### Step 1: Calculate \( \frac{dx}{dt} \)

Given:

\[ x = \frac{1+t}{t^3} \]

We can differentiate \( x \) with respect to \( t \):

\[ \frac{dx}{dt} = \frac{d}{dt} \left( \frac{1+t}{t^3} \right) \]

Using the quotient rule:

\[ \frac{dx}{dt} = \frac{(t^3)(1) - (1+t)(3t^2)}{(t^3)^2} \]

Simplifying the numerator:

\[ = \frac{t^3 - (3t^2 + 3t^3)}{t^6} = \frac{-2t^3 - 3t^2}{t^6} = \frac{-2t - 3}{t^4} \]

### Step 2: Calculate \( \frac{dy}{dt} \)

Given:

\[ y = \frac{3}{2t^2} + \frac{2}{t} \]

Differentiating \( y \) with respect to \( t \):

\[ \frac{dy}{dt} = \frac{d}{dt} \left( \frac{3}{2t^2} \right) + \frac{d}{dt} \left( \frac{2}{t} \right) \]

Calculating each term:

\[ \frac{dy}{dt} = -\frac{3}{t^3} - \frac{2}{t^2} \]

Combining the terms:

\[ = -\frac{3 + 2t}{t^3} \]

### Step 3: Calculate \( \frac{dy}{dx} \)

Now we can find \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-\frac{3 + 2t}{t^3}}{\frac{-2t - 3}{t^4}} = \frac{(3 + 2t) t^4}{(2t + 3) t^3} = \frac{(3 + 2t) t}{(2t + 3)} \]

### Step 4: Substitute \( \frac{dy}{dx} \) into the given equation

Now substituting \( \frac{dy}{dx} \) into the equation:

\[ f(x) \left( \frac{(3 + 2t)t}{(2t + 3)} \right)^3 = 1 + \frac{(3 + 2t)t}{(2t + 3)} \]

Let \( \frac{(3 + 2t)t}{(2t + 3)} = k \):

\[ f(x) k^3 = 1 + k \]

### Step 5: Solve for \( f(x) \)

Rearranging gives:

\[ f(x) = \frac{1 + k}{k^3} \]

### Step 6: Substitute back for \( k \)

Substituting back for \( k \):

\[ k = \frac{(3 + 2t)t}{(2t + 3)} \]

Thus:

\[ f(x) = \frac{1 + \frac{(3 + 2t)t}{(2t + 3)}}{\left( \frac{(3 + 2t)t}{(2t + 3)} \right)^3} \]

### Step 7: Simplify \( f(x) \)

After simplification, we find that:

\[ f(x) = x \]

### Final Answer

Thus, we conclude that:

\[ f(x) = x \]