Difference between first and second fundamental theorem of calculus

In first fundamental theorem of calculus,it states if $A(x)={\int}_{a}^{x}f(t)dt$ then ${A}^{\prime}(x)=f(x)$.But in second they say ${\int}_{a}^{b}f(t)dt=F(b)-F(a)$,But if we put x=b in the first one we get A(b).Then what is the difference between these two and how do we prove A(b)=F(b)−F(a)?

In first fundamental theorem of calculus,it states if $A(x)={\int}_{a}^{x}f(t)dt$ then ${A}^{\prime}(x)=f(x)$.But in second they say ${\int}_{a}^{b}f(t)dt=F(b)-F(a)$,But if we put x=b in the first one we get A(b).Then what is the difference between these two and how do we prove A(b)=F(b)−F(a)?