Hello, UMStudent!

This is an exercise in handling the Chain Rule.

Suppose: .F(x) = p(x^9) .and .G(x) = [p(x)]^9

Also suppose: .a^8 = 6, .p(a) = 2, .p'(a) = 13, .p'(a^9) - 15

Find: .(a) F'(a) .and .(b) G'(a)

(a) We are given: .F(x) .= .p(x^9)

Differentiate: .F'(x) .= .p'(x^9) · 9x^8

Then we have: .F(a) .= .p'(a^9) · 9a^8

. . . . . . . . . . . . . . . . . . . .↓ . . . . ↓

Substitute: . . . F(a) .= . . .15 · 9 · 6 . = . 810

(b) We are given: .G(x) .= .[p(x)]^9

Differentiate: .G'(x) .= .9[p(x)]^8 · p'(x)

Then we have: .G'(a) .= .9[p(a)]^8 · p'(a)

. . . . . . . . . . . . . . . . . . . . . ↓ . . . . ↓

Substitute: . . . G'(a) .= .9 · (2^8) · 13 . = . 29,952