Producing Faraday's law in differential form

Faraday's law in intergral form:

(1)
Applying Stoke's Thm:

(2)
yields:

(3)
Replacing the right hand side, using Eqn. 1:

(4)
Now, from:

(5)
over the same area, S:

(6)

(7)
we get Faraday's law in differential form:

(8)