If : `3f(x)-2f((1)/(x))=x," then: "f'(2)=`

Options

A. `2//7`
B. `1//2`
C. 2
D. `7//2`

Answer

To solve the problem, we start with the equation given:

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

### Step 1: Differentiate both sides with respect to \( x \)

Using the differentiation rules, we differentiate the left-hand side:

\[ \frac{d}{dx}[3f(x)] - \frac{d}{dx}[2f\left(\frac{1}{x}\right)] = \frac{d}{dx}[x] \]

This gives us:

\[ 3f'(x) - 2 \cdot f'\left(\frac{1}{x}\right) \cdot \left(-\frac{1}{x^2}\right) = 1 \]

### Step 2: Simplify the differentiated equation

Rearranging the equation, we have:

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

### Step 3: Substitute \( x = 2 \)

We want to find \( f'(2) \), so we substitute \( x = 2 \):

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

This simplifies to:

\[ 3f'(2) + \frac{2}{4} f'\left(\frac{1}{2}\right) = 1 \]

or

\[ 3f'(2) + \frac{1}{2} f'\left(\frac{1}{2}\right) = 1 \] (Equation 1)

### Step 4: Substitute \( x = \frac{1}{2} \)

Next, we substitute \( x = \frac{1}{2} \) into the differentiated equation:

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

This simplifies to:

\[ 3f'\left(\frac{1}{2}\right) + 8f'(2) = 1 \] (Equation 2)

### Step 5: Solve the system of equations

Now we have two equations:

1. \( 3f'(2) + \frac{1}{2} f'\left(\frac{1}{2}\right) = 1 \)
2. \( 3f'\left(\frac{1}{2}\right) + 8f'(2) = 1 \)

From Equation 1, we can express \( f'\left(\frac{1}{2}\right) \):

\[ \frac{1}{2} f'\left(\frac{1}{2}\right) = 1 - 3f'(2) \]

Multiplying through by 2:

\[ f'\left(\frac{1}{2}\right) = 2 - 6f'(2) \]

### Step 6: Substitute back into Equation 2

Substituting this expression into Equation 2:

\[ 3(2 - 6f'(2)) + 8f'(2) = 1 \]

Expanding gives:

\[ 6 - 18f'(2) + 8f'(2) = 1 \]

Combining like terms:

\[ 6 - 10f'(2) = 1 \]

### Step 7: Solve for \( f'(2) \)

Rearranging gives:

\[ -10f'(2) = 1 - 6 \]

\[ -10f'(2) = -5 \]

Dividing by -10:

\[ f'(2) = \frac{1}{2} \]

### Final Answer

Thus, the value of \( f'(2) \) is:

\[ \boxed{\frac{1}{2}} \]