Find the derivative of f(x) = 4 divided by x at x = 2.

Hello!The answer is:[tex]f'(2)=-1[/tex]Why?To solve this problem, first we need to derivate the given function, and then, evaluate the derivated function with x equal to 2.The given function is:[tex]f(x)=\frac{4}{x}[/tex]It's a quotient, so, we need to use the following formula to derivate it:[tex]f'(x)=\frac{d}{dx}(\frac{u}{v}) =\frac{v*u'-u*v'}{v^{2} }[/tex]Then, of the given function we have that:[tex]u=4\\v=x[/tex]So, derivating we have:[tex]f'(x)=\frac{d}{dx}(\frac{4}{x}) =\frac{x*(4)'-4*(x)'}{x^{2} }[/tex][tex]f'(x)=\frac{d}{dx}(\frac{4}{x}) =\frac{x*0-4*1}{x^{2} }[/tex][tex]f'(x)=\frac{d}{dx}(\frac{4}{x}) =\frac{0-4}{x^{2} }[/tex][tex]f'(x)=\frac{d}{dx}(\frac{4}{x}) =\frac{-4}{x^{2} }[/tex]Hence,[tex]f'(x)=\frac{d}{dx}(\frac{4}{x}) =\frac{-4}{x^{2} }[/tex]Now, evaluating with x equal to 2, we have:[tex]f'(2)=\frac{-4}{(2)^{2} }[/tex][tex]f'(2)=\frac{-4}{4}[/tex][tex]f'(2)=-1[/tex]Therefore, the answer is:[tex]f'(2)=-1[/tex]Have a nice day!