As we have seen, the height h(t) of the ball thrown vertically obeys:

h(t) = 32t -16t²

We calculate the derivative of h at t₀

step1

Factor out t - t₀

h(t) - h(t₀) = 32t -16t2 - (32t₀ - 16 t₀²)

           = 32(t - t₀) - 16 (t² - t₀²)

                  = 32(t - t₀) - 16 (t - t₀)(t + t₀)

         = (t - t₀)(32 - 16(t + t₀))

step 2

Divide by t - t₀

 = 32 - 16(t + t₀)

step 3

Take the limit

h'(t₀) = = 32 - 32t₀