Plot[Cos[x] - x, {x, -10, 10}, PlotStyle -> {Blue, Thick}, PlotRange ->{-10, 10}, AspectRatio ->Automatic, Ticks -> {Range[-10,10], Range[-10,10]}]

Plot[Cos[x] - x, {x, -1.3, 1.3}, PlotStyle -> {Blue, Thick}, PlotRange ->{-1.1, 1.1}, AspectRatio ->Automatic, Ticks -> {Range[-1.2,1.2,.2], Range[-1,1,.2]}]

Plot[{Cos[x] - x, -x+1}, {x, -1.3, 1.3}, PlotStyle -> {{Blue, Thick}, {Red, Thick}}, PlotRange ->{-1.1, 1.1}, AspectRatio ->Automatic, Ticks -> {Range[-1.2,1.2,.2], Range[-1,1,.2]}]

graphic1 = Plot[{Cos[x] - x, -x+1}, {x, .6, 1.2}, PlotStyle -> {{Blue, Thick}, {Red, Thick}}, PlotRange ->{-.5, .2}, AspectRatio ->Automatic, Ticks -> {Range[.6,1.2,.05], Range[-1,1,.05]}]

graphic2 = ParametricPlot[{1, -x},  {x, 0, .455}, PlotStyle -> {Darker[Green], Dashing[.015], Thick}, PlotRange ->{{.6, 1.2}, {-.5, .2}}, AspectRatio ->Automatic, Ticks -> {Range[.6,1.2,.05], Range[-1,1,.05]}]

Show[graphic1,graphic2]

graphic3 =  Plot[{Cos[x] - x, f'[1](x-1) + f[1]}, {x, .6, 1.2}, PlotStyle -> {{Blue, Thick}, {Darker[Green], Thick}}, PlotRange ->{-.5, .2}, AspectRatio ->Automatic, Ticks -> {Range[.6,1.2,.05], Range[-1,1,.05]}]

Show[graphic1,graphic2,graphic3]

f[x_] = Cos[x] - x

graphic5 =  Plot[{Cos[x] - x, f'[1](x-1) + f[1]}, {x, .72, .78}, PlotStyle -> {{Blue, Thick}, {Darker[Green], Thick}}, PlotRange ->{-.04, .04}, AspectRatio ->Automatic, Ticks -> {Range[.72,.78,.005], Range[-.04,.04,.005]}]

graphic4 = ParametricPlot[{.75, -x},  {x, 0, .018}, PlotStyle -> {Orange, Dashing[.015], Thick}, PlotRange ->{{.72,.78}, {-.04,.04}}, AspectRatio ->Automatic, Ticks -> {Range[.72,.78,.005], Range[-.04,.04,.005]}]

Show[graphic4, graphic5]

graphic6 =  Plot[{Cos[x] - x, f'[.75](x-.75) + f[.75]}, {x, .72, .78}, PlotStyle -> {{Blue, Thick}, {Orange, Thick}}, PlotRange ->{-.04, .04}, AspectRatio ->Automatic, Ticks -> {Range[.72,.78,.005], Range[-.04,.04,.005]}]

Show[graphic4, graphic5,graphic6]

graphic6 =  Plot[{Cos[x] - x, f'[.75](x-.75) + f[.75]}, {x, .7385, .7395}, PlotStyle -> {{Blue, Thick}, {Orange,Thick}}, PlotRange ->{-.0005, .0005}, AspectRatio ->Automatic, Ticks -> {Range[.7385,.7395,.00025], Range[-.001,.001,.0002]}]