Usa la ecuación 3.5: r′(t)=limΔt→0r(t+Δt)−r(t)Δt=limΔt→0[(3(t+Δt)+4)i+((t+Δt)2−4(t+Δt)+3)j]−[(3t+4)i+(t2−4t+3)j]Δt=limΔt→0(3Δt)i+(2tΔt+(Δt)2−4Δt)jΔt=limΔt→0(3i+(2t+Δt−4)j)=3i+(2t−4)j\begin{aligned} \bold{r'}(t) &= \lim\limits_{\Delta t \to 0}\frac{\bold{r}(t+\Delta t)-\bold{r}(t)}{\Delta t}\\ &= \lim\limits_{\Delta t \to 0}\frac{[(3(t+\Delta t)+4)\bold{i} + ((t+\Delta t)^2−4(t+\Delta t)+3)\bold{j}]−[(3t+4)\bold{i}+(t^2−4t+3)\bold{j}]}{\Delta t}\\ &= \lim\limits_{\Delta t \to 0}\frac{(3\Delta t)\bold{i}+(2t\Delta t +(\Delta t)^2−4\Delta t)\bold{j}}{\Delta t}\\ &= \lim\limits_{\Delta t \to 0}(3\bold{i}+(2t+\Delta t−4)\bold{j})\\ &= 3\bold{i}+(2t−4)\bold{j} \end{aligned}r′(t)=Δt→0limΔtr(t+Δt)−r(t)=Δt→0limΔt[(3(t+Δt)+4)i+((t+Δt)2−4(t+Δt)+3)j]−[(3t+4)i+(t2−4t+3)j]=Δt→0limΔt(3Δt)i+(2tΔt+(Δt)2−4Δt)j=Δt→0lim(3i+(2t+Δt−4)j)=3i+(2t−4)j