Find there derivative

please show work and don't just list the rules,


(a^2+3a+5)(6a^2+4)

f(a) =
(a^2 + 3a + 5) * (6a^2 + 4) =
6a^4 + 4a^2 + 18a^3 + 12a + 30a^2 + 20 =
6a^4 + 18a^3 + 34a^2 + 12a + 20



f(a + h) =
((a + h)^2 + 3 * (a + h) + 5) * (6 * (a + h)^2 + 4) =
(6 * (a + h)^4 + 4 * (a + h)^2 + 18 * (a + h)^3 + 12 * (a + h) + 30 * (a + h)^2 + 20) =>
6 * (a + h)^4 + 18 * (a + h)^3 + 34 * (a + h)^2 + 12 * (a + h) + 20



f(a + h) - f(a) =

6 * (a + 4)^4 + 18 * (a + h)^3 + 34 * (a + h)^2 + 12 * (a + h) + 20 - 6 * a^4 - 18 * a^3 - 34 * a^2 - 12 * a - 20 =

6 * ((a + h)^4 - a^4) + 18 * ((a + h)^3 - a^3) + 34 * ((a + h)^2 - a^2) + 12 * ((a + h) - a) + 20 - 20 =>

6 * (a^4 + 4a^3 * h + 6a^2 * h^2 + 4a * h^3 + h^4 - a^4) + 18 * (a^3 + 3a^2 * h + 3a * h^2 + h^3 - a^3) + 34 * (a^2 + 2ah + h^2 - a^2) + 12 * (h) =>

6 * (4a^3 * h + 6a^2 * h^2 + 4a * h^3 + h^4) + 18 * (3a^2 * h + 3a * h^2 + h^3) + 34 * (2ah + h^2) + 12h =>

6 * h * (4a^3 + 6a^2 * h + 4a * h^2 + h^3) + 18 * h * (3a^2 + 3a * h + h^2) + 34 * h * (2a + h) + 12h

Divide that all by h

6 * (4a^3 + 6a^2 * h + 4a * h^2 + h^3) + 18 * (3a^2 + 3ah + h^2) + 34 * (2a + h) + 12

Let h go to 0

6 * (4a^3) + 18 * (3a^2) + 34 * 2a + 12

24a^3 + 54a^2 + 68a + 12


Or we could have just used the product rule and the power rule:

u = a^2 + 3a + 5
du = 2a + 3
v = 6a^2 + 4
dv = 12a

d(u * v) = udv + vdu

(a^2 + 3a + 5) * 12a + (6a^2 + 4) * (2a + 3) =>
12a^3 + 36a^2 + 60a + 12a^3 + 18a^2 + 8a + 12 =>
12a^3 + 12a^3 + 36a^2 + 18a^2 + 60a + 8a + 12 =>
24a^3 + 54a^2 + 68a + 12

f(x) = (a^2+3a+5)(6a^2+4)

= 6a^4 + 4a^2 + 18a^3 + 12a + 30a^2 + 20

f '(x) = 24a^3 + 8a + 54a^2 + 12 + 60a

= 24a^3 + 54a^2 +68a + 12