Please help find the derivative and tell what rules you used to solve

Question

g(t)=(t3-pi³)²(t²+pi²)³

Thank you!

Also, the directions did not state how many decimals into pi to solve for

Answer 1

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Let P = pi; leave the pi ALONE... don't use decimals

Pascal's Triangle:

1

1 1

1 2 1

1 3 3 1

(A+B)^3 = A^3 + 3 A^2 B + 3 A B^2 + B^3

A = t^2 and B = P^2

G(t) = (t^3 - P^3)^2 ( t^2 + P^2)^3

= (t^6 - 2 t^3 P^3 + P^6) (t^6 + 3t^4 P^2 + 3 t^2 P^4 + P^6) <-- FOIL & Pascal's Triangle

= t^12 + 3t^10 P^2 - 2 t^9 p^3 + 3 t^8 p^4 - 6 t^7 p^5 + 2t^6 p^6 - 6t^5 p^7

+ 3t^4 p^8 - 2t^3 p^9 + 3t^2 p^10 + p^12

dG/dt = G'(t) = 12 t^11 + 30 t^9 p^2 - 18 t^8p^3 + 24 t^7 p^4 -42t^6 p^5 + 12t^5 p^6 - 30t^4p^7

+12 t^3 p&8 - 6t^2p^9 + 6 t p^10

(Please label this Derivative version A)

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f(t) = t^3 - P^3 --> f' = 3t^2 by chain rule

h(t) = t^2 + p^2 --> h' = 2t by chain rule

Then G(t) = f^2 * h^3

By product rule: dG/dt = G'(t) = 3 f^2 h^2 h' + 2 f h^3 f'

= f h^2 ( 3 f h' + 2 h f')

= (t^3 - p^3) (t^2 + p^2)^2 [ 3(t^3 - p^3)(2t) + 2(t^2 + P^2)(3t^2)]

= (t^3 - p^3) (t^2 + p^2)^2 (6t) [ t^3 - p^3 + t(t^2 + p^2)]

= (t^3 - p^3) (t^2 + p^2)^2 (6t) [ t^3 - p^3 + t^3 + t*p^2]

= (t^3 - p^3) (t^2 + p^2)^2 (6t) [ 2t^3 - p^3 + t*p^2]

= (t^3 - p^3) (t^4 + 2 t^2 p^2 + p^4) [ 12t^4 - 6 t p^3 + 6t^2 p^2]

= (t^7 + 2 t^5 p^2 + t^3 p^4 - t^4 p^3 - 2 t^2 p^5 - p^7)[ 12t^4 - 6 t p^3 + 6t^2 p^2]

= 12 t^11 - 6 t^8 p^3 + 6 t^9 p^2 <--- multiplies t^7 by 2nd polynomial

24 t^9 p^2 - 12 t^6 p^5 + 12 t^7 p^4 <--- multiplies 2 t^5 p^2 by 2nd polynomial

12 t^7 p^4 - 6t^4 p^7 + 6t^5 p^6 <--- multiplies t^3 p^4 by 2nd polynomial

- 12 t^8 p^3 + 6t^5 p^6 - 6 t^6 p^5 <--- multiplies -t^4 p^3 by 2nd polynomial

-24t^6 p^5 + 12t^3p^8 - 12t^4 p^7 <--- multiplies -2 t^2 p^5 by 2nd polynomial

-12 t^4 p^7 +6 t p^10 - 6 t^2 p^9 <--- multiplies -p^7 by 2nd polynomial

shall be derivative version B,

which, when combined, is exactly equal to derivative version A above...

So these two derivatives are the same and agree when everything in the function is multiplied,

versus using the product rule, which PROVES the answer is correct.

Final answer:

dG/dt = G'(t) = 12 t^11 + 30 t^9 p^2 - 18 t^8p^3 + 24 t^7 p^4 -42t^6 p^5 + 12t^5 p^6 - 30t^4p^7

+12 t^3 p^8 - 6t^2p^9 + 6 t p^10