Final answer:
To find the equation of the tangent line at a given point, we calculate the derivative of the function for the slope and then use the point-slope form with the calculated slope and the given point.
Step-by-step explanation:
To find an equation for the tangent line to the graph of the function f(x) = 6x/(x+6) at the point (1, 0.857142857142857), we first need to determine the slope of the tangent line at that point. This is done by finding the derivative of the function, which represents the slope of the tangent line at any point on the curve.
First, let's find the derivative of the function f(x). Since f(x) is a quotient, we will use the quotient rule: f'(x) = (v*g'(u) - u*g'(v)) / v^2, where u is the numerator and v is the denominator. After finding the derivative, we will evaluate it at x = 1 to get the slope of the tangent line at the given point.
Next, we use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point on the line (in this case, (1, 0.857142857142857)), and m is the slope we found. Substituting the slope and the point into this equation, we get the equation of the tangent line.