We start by calculating Δf: Δf=f(x+Δx)−f(x)=(x+Δx)2−x2=x2+2xΔx+Δx2−x2=2xΔx+Δx2 We now calcu...We start by calculating Δf: Δf=f(x+Δx)−f(x)=(x+Δx)2−x2=x2+2xΔx+Δx2−x2=2xΔx+Δx2 We now calculate ΔfΔx: ΔfΔx=2xΔx+Δx2Δx=2x+Δx and take the limit Δx→0: \[\begin{aligned} \frac{df}{dx}&=\lim_{\Delta x\to 0 }\frac{\Delta f}{\Delta x}\\ &=\lim_{\Delta x\to 0 }(2x+\…