Find the equation of the tangent line to the curve \( y=\sqrt{x}+\frac{1}{\sqrt{x}} \) at point \( \left(4, \frac{5}{2}\right) \) \[ \text { ans: } y=\frac{3}{16} x+\frac{7}{4} \]

Given the curve \[ y = \sqrt{x} + \frac{1}{\sqrt{x}} = x^{\frac{1}{2}} + x^{-\frac{1}{2}}, \] we first find its derivative. 1. **Differentiate the function:** The derivative of \( x^{\frac{1}{2}} \) is \[ \frac{d}{dx}x^{\frac{1}{2}} = \frac{1}{2}x^{-\frac{1}{2}}, \] and the derivative of \( x^{-\frac{1}{2}} \) is \[ \frac{d}{dx}x^{-\frac{1}{2}} = -\frac{1}{2}x^{-\frac{3}{2}}. \] Thus, the derivative of \( y \) is \[ y' = \frac{1}{2}x^{-\frac{1}{2}} - \frac{1}{2}x^{-\frac{3}{2}}. \] 2. **Evaluate the derivative at \( x = 4 \):** We compute \( x^{-\frac{1}{2}} \) and \( x^{-\frac{3}{2}} \) at \( x = 4 \): \[ 4^{-\frac{1}{2}} = \frac{1}{\sqrt{4}} = \frac{1}{2}, \] \[ 4^{-\frac{3}{2}} = \frac{1}{4^{\frac{3}{2}}} = \frac{1}{(\sqrt{4})^3} = \frac{1}{8}. \] Substituting into \( y' \): \[ y'(4) = \frac{1}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \frac{1}{8} = \frac{1}{4} - \frac{1}{16}. \] To combine the fractions, use a common denominator of 16: \[ \frac{1}{4} = \frac{4}{16}, \quad y'(4) = \frac{4}{16} - \frac{1}{16} = \frac{3}{16}. \] This means the slope \( m \) of the tangent line at the point \( (4, \frac{5}{2}) \) is \[ m = \frac{3}{16}. \] 3. **Write the equation of the tangent line:** Using the point-slope form \[ y - y_1 = m(x - x_1), \] where \( (x_1, y_1) = \left(4, \frac{5}{2}\right) \) and \( m = \frac{3}{16} \), we have: \[ y - \frac{5}{2} = \frac{3}{16}(x - 4). \] 4. **Simplify to slope-intercept form:** Distribute the slope: \[ y - \frac{5}{2} = \frac{3}{16}x - \frac{3}{16} \times 4 = \frac{3}{16}x - \frac{12}{16} = \frac{3}{16}x - \frac{3}{4}. \] Add \( \frac{5}{2} \) to both sides: \[ y = \frac{3}{16}x - \frac{3}{4} + \frac{5}{2}. \] Express \( \frac{5}{2} \) with a denominator of 4: \[ \frac{5}{2} = \frac{10}{4}, \] then combine: \[ -\frac{3}{4} + \frac{10}{4} = \frac{7}{4}. \] The final equation of the tangent line is: \[ y = \frac{3}{16}x + \frac{7}{4}. \]