Question

find f'(g(x)) using the chain rule.
y = (1+cot^5(x^4+1))^9

Answer 1

I'm confused... we'd use the chain rule multiple times, what "f'(g(x))" is it referring to...?

Answer 2

its a chain within a chain within a chain...

Answer 3

y = (1+cot^5(x^4+1))^9
y = (1 + (cot(x^4 + 1))^5)^9

y' = 9(1 + (cot(x^4 + 1))^5)^8 * (0 + 5(cot(x^4 + 1))^4) * (-(csc(x^4 + 1))^2) * (4x^3)
= -180x^3 (csc(x^4 + 1))^2 (cot(x^4 + 1))^4 (1 + (cot(x^4 + 1))^5)^8

A)
f(x) = x^9
g(x) = 1 + (cot(x^4 + 1))^5
f'(x) = 9x^8
f'(g(x)) = 9(1 + (cot(x^4 + 1))^5)^8

B)
f(x) = x^5
g(x) = cot(x^4 + 1)
f'(x) = 5x^4
f'(g(x)) = 5(cot(x^4 + 1))^4

C)
f(x) = cot(x)
g(x) = x^4 + 1
f'(x) = -(csc(x))^2
f'(g(x)) = -(csc(x^4 + 1))^2