{x0,x1,x2,x3,x4}={0,3,5,8,13}
{d0,d1,d2,d3,d4}={0,225,383,623,993}
{s0,s1,s2,s3,s4}={75,77,80,74,72}

FD[f2_,f1_,x2_,x1_]:=(f2-f1)/(x2-x1);

tvals={x0,x0,x1,x1,x2,x2,x3,x3,x4,x4}
fvals={d0,d0,d1,d1,d2,d2,d3,d3,d4, d4}
firsts={
s0,
FD[fvals[[3]],fvals[[2]],tvals[[3]],tvals[[2]]],
s1,
FD[fvals[[5]],fvals[[4]],tvals[[5]],tvals[[4]]],
s2,
FD[fvals[[7]],fvals[[6]],tvals[[7]],tvals[[6]]],
s3,
FD[fvals[[9]],fvals[[8]],tvals[[9]],tvals[[8]]],
s4}

n=8
seconds=Table[FD[firsts[[i+1]],firsts[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]
n=n-1
thirds=Table[FD[seconds[[i+1]],seconds[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]
n=n-1
fourths=Table[FD[thirds[[i+1]],thirds[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]
n=n-1
fifths=Table[FD[fourths[[i+1]],fourths[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]
n=n-1
sixths=Table[FD[fifths[[i+1]],fifths[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]
n=n-1
sevenths=Table[FD[sixths[[i+1]],sixths[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]
n=n-1
eighths=Table[FD[sevenths[[i+1]],sevenths[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]
n=n-1
ninths=Table[FD[eighths[[i+1]],eighths[[i]],tvals[[i+10-n]],tvals[[i]]],{i,1,n}]

p[x_]=fvals[[1]]+firsts[[1]]*(x-x0)+seconds[[1]]*(x-x0)^2+thirds[[1]]*(x-x0)^2*(x-x1)+fourths[[1]]*(x-x0)^2*(x-x1)^2+fifths[[1]]*(x-x0)^2*(x-x1)^2*(x-x2)+sixths[[1]]*(x-x0)^2*(x-x1)^2*(x-x2)^2+sevenths[[1]]*(x-x0)^2*(x-x1)^2*(x-x2)^2*(x-x3)+eighths[[1]]*(x-x0)^2*(x-x1)^2*(x-x2)^2*(x-x3)^2+ninths[[1]]*(x-x0)^2*(x-x1)^2*(x-x2)^2*(x-x3)^2*(x-x4)
Plot[{p[x]},{x,x0,x4},PlotStyle->{Blue}]

dpdx[x_]=D[p[x],{x,1}];
d2pdx2[x_]=D[p[x],{x,2}];

p[10]*1.0
dpdx[10]*1.0
Plot[{dpdx[x]},{x,x0,x4},PlotStyle->{Red}]

NSolve[dpdx[x]==80+2/3,x]

crit=NSolve[d2pdx2[x]==0,x]
mphConversion=3600./5280.
dpdx[12.3718]*mphConversion