Question

# f(x)=2x(f(x+Δx)-f(x))/Δx

Transformations of functions

$$\displaystyle{f{{\left({x}\right)}}}={2}{x}$$
$$\displaystyle\frac{{ f{{\left({x}+\triangle{x}\right)}}- f{{\left({x}\right)}}}}{\triangle}{x}$$

Using the given function, substitute to the difference quotient and simplify: $$(f(x+\triangle x)-f(x))/\triangle x=(2(x+\triangle x)-2x)/\triangle x\\ \displaystyle=\frac{{{2}{x}+{2}\triangle{x}-{2}{x}}}{\triangle}{x}\\ \displaystyle={2}\triangle\frac{x}{\triangle}{x}\\ =2$$