Question:

Given the functional equation: $$3f(x) - 2f\left(\frac{1}{x}\right) = x \quad (*)$$ Step 1: Substitute $x$ with $\frac{1}{x}$ in equation $(*)$. $$3f\left(\frac{1}{x}\right) - 2f(x) = \frac{1}{x} \quad (**)$$ Step 2: Solve the system of equations $(*)$ and $(**)$ for $f(x)$. Multiply equation $(*)$ by 3: $$9f(x) - 6f\left(\frac{1}{x}\right) = 3x \quad (1)$$ Multiply equation $(**)$ by 2: $$6f\left(\frac{1}{x}\right) - 4f(x) = \frac{2}{x} \quad (2)$$ Add equation (1) and equation (2): $$(9f(x) - 6f\left(\frac{1}{x}\right)) + (6f\left(\frac{1}{x}\right) - 4f(x)) = 3x + \frac{2}{x}$$ $$5f(x) = 3x + \frac{2}{x}$$ $$f(x) = \frac{3x}{5} + \frac{2}{5x}$$ Step 3: Differentiate $f(x)$ with respect to $x$. $$f'(x) = \frac{d}{dx}\left(\frac{3x}{5} + \frac{2}{5x}\right)$$ $$f'(x) = \frac{3}{5} - \frac{2}{5x^2}$$ Step 4: Evaluate $f'(2)$. $$f'(2) = \frac{3}{5} - \frac{2}{5(2^2)}$$ $$f'(2) = \frac{3}{5} - \frac{2}{5 \cdot 4}$$ $$f'(2) = \frac{3}{5} - \frac{2}{20}$$ $$f'(2) = \frac{3}{5} - \frac{1}{10}$$ To combine these terms, find a common denominator: $$f'(2) = \frac{6}{10} - \frac{1}{10}$$ $$f'(2) = \frac{5}{10}$$ $$f'(2) = \frac{1}{2}$$