Scratch

f[x_]:=(x-1)(x-3)(x-4)(x-5)

f=Function[x,(x-1) (x-3) (x-4) (x-5)]

D[f[x],x]

Plot[f[x],{x,0,6}]

Plot[D[f[x],x]/.{x->a},{a,0,6},PlotStyle->Red]

Individual

Manipulate[Show[{Plot[f[x],{x,0,6}],Plot[(D[f[y],y]/.{y->a})(x-a)+f[a],{x,0,6},PlotStyle->Red],Graphics[{PointSize[Large],Point[{a,f[a]}]}]}],{a,0,6}]

Manipulate[Show[{Plot[D[f[x],x]/.{x->a},{a,0,6},PlotStyle->Red],Graphics[{PointSize[Large],Point[{a,(D[f[y],y]/.{y->a})}]}]}],{a,0,6}]

Together

Manipulate[Row[{Show[{Plot[f[x],{x,0,6}],Plot[(D[f[y],y]/.{y->a})(x-a)+f[a],{x,0,6},PlotStyle->Red],Graphics[{PointSize[Large],Point[{a,f[a]}]}]},ImageSize->300],Show[{Plot[D[f[x],x]/.{x->a},{a,0,6},PlotStyle->Red],Graphics[{PointSize[Large],Point[{a,(D[f[y],y]/.{y->a})}]}]},ImageSize->300]}],{a,0,6}]