Final answer:
To find the equation of the tangent line to the graph of f(x) = 1/√x at x = 9, calculate the derivative to get the slope, evaluate at x = 9 for the slope and function, and use the point-slope form with these values to write the equation.
Step-by-step explanation:
To find an equation of the tangent line to the graph of f(x) = 1/√x at x=9, we need to calculate the derivative of the function, which represents the slope of the tangent line, and evaluate it at x = 9. The derivative of f(x) = 1/√x is f'(x) = -1/(2x³²). When x = 9, f'(9) = -1/(2*9³²) = -1/54. Therefore, the slope of the tangent line at x = 9 is -1/54.
Next, we find the y-coordinate of the point on the curve where x = 9 by evaluating the function, giving us f(9) = 1/√9 = 1/3. This point is (9, 1/3).
The equation of the tangent line can then be expressed using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the point on the curve and m is the slope. Substituting the values, we get the equation of the tangent line: y - 1/3 = -1/54(x - 9).