Final answer:
The derivative of the function f(x) = 7x/(x-1) is found using the quotient rule and simplifying the result, which gives us f'(x) = -7/((x-1)^2).
Step-by-step explanation:
To find the derivative of the function f(x) = 7x/(x-1) using the alternative formula for derivatives, we will use the quotient rule. The quotient rule states that if you have a function that is the ratio of two functions, u(x) and v(x), the derivative of that function is given by: f'(x) = (v(x)u'(x) - u(x)v'(x)) / (v(x))^2. In our case, u(x) = 7x and v(x) = x-1. Differentiating both u(x) and v(x) gives us u'(x) = 7 and v'(x) = 1.
The derivative of the function f(x) can be found by substituting the derivatives of u and v into the quotient rule formula.
f'(x) = ((x-1)*7 - 7x*(1))/((x-1)^2)
Simplifying, we get f'(x) = (7x-7 - 7x)/((x-1)^2) = -7/((x-1)^2).