Is formula #7 in this MIT OCW course incorrect? I think it should be $$f(x)=(\sqrt{1-(mx-c)^2}+d)/m$$
Also, presumably this answer to a very similar problem is also wrong. Because this is not the correct solution to the differential equation
$$\lambda-\frac{1}{\big(1+(y')^2\big)^{3/2}}=0$$
The linked MSE answer, which purports that the shortest curve is a parabola, is incorrect.
The linked OCW material is correct. Indeed, $$\frac{d}{dx} \left( -\frac{1}{m} \sqrt{1-(mx-c)^2} \right) = \frac{mx-c}{\sqrt{1-(mx-c)^2}} $$ You dropped the minus sign somewhere.