For the equation \( x^{\frac{3}{5}}+y^{\frac{1}{2}}=7 \). Find the equation of the tangent line at the point \( (1,36) \). (Use symbolic notation and fractions where needed.) Equation of the tangent line is \( y=\square \) help (fractions)

To find the equation of the tangent line at the point \( (1, 36) \) for the equation \( x^{\frac{3}{5}} + y^{\frac{1}{2}} = 7 \), we first need to differentiate the equation implicitly. 1. **Differentiate**: \[ \frac{3}{5}x^{-\frac{2}{5}} + \frac{1}{2}y^{-\frac{1}{2}}\frac{dy}{dx} = 0. \] 2. **Solve for** \( \frac{dy}{dx} \): \[ \frac{1}{2}y^{-\frac{1}{2}}\frac{dy}{dx} = -\frac{3}{5}x^{-\frac{2}{5}}, \] \[ \frac{dy}{dx} = -\frac{3}{5} \cdot \frac{2y^{\frac{1}{2}}}{x^{\frac{2}{5}}}. \] 3. **Substitute the point** \( (1, 36) \) into the derivative: \[ \frac{dy}{dx} = -\frac{3}{5} \cdot \frac{2(36)^{\frac{1}{2}}}{(1)^{\frac{2}{5}}} = -\frac{3}{5} \cdot \frac{2(6)}{1} = -\frac{36}{5}. \] 4. **Use the point-slope form** of the equation of a line: \[ y - 36 = -\frac{36}{5}(x - 1). \] 5. **Rearranging this gives**: \[ y = -\frac{36}{5}x + \frac{36}{5} + 36, \] \[ y = -\frac{36}{5}x + \frac{36 + 180}{5}, \] \[ y = -\frac{36}{5}x + \frac{216}{5}. \] Thus, the equation of the tangent line is: \[ \boxed{y = -\frac{36}{5}x + \frac{216}{5}}. \]