Chain Rule. Is this correct?

Question

y(x)=(9x^2 + 4x)^1/3

If u= 9x^2+4x

du/dx= 18x+4 and dy/dx= (1/3)U^-2/3=1/3(9x^2+4x)^(-2/3)

is the final answer 6x + (4/3)(9x^2 + 4x)^(-2/3)?

Answer 1

Another way is to say"differentiate the bracket and then differentiate what is inside the bracket."
f `(x) = [ (1/3) (9x² + 4x)^(- 2/3) ] [18x + 4]
f `(x) = (6x + 4 / 3) (9x² + 4x)^(- 2 / 3)

Answer 2

Almost. You just didn't distribute the du/dx term correctly in the end.

dy/dx = (1/3)u^(-2/3) * du/dx, so
dy/dx = (1/3)(9x^2 + 4x)^(-2/3) * (18x + 4)

This is also
6(9x^2 + 4x)^(-2/3) + (4/3)(9x^2 + 4x)^(-2/3)

Answer 3

du^n / dx = n (u)^(n - 1) d/dx u


y'(x) = (9x^2 + 4x)^(1/3)

y'(x) = 1/3 * (9x^2 + 4x)^(1/3 - 1) d/dx (9x^2 + 4x)

y'(x) = d/dx (9x^2 + 4x) / [ 3 (9x^2 + 4x)^(2/3) ]

y'(x) = (18x + 4) / [ 3 (9x^2 + 4x)^(2/3) ]

Answer 4

the final answer should be

y'(x) = 24x ( 9x^2 + 4x )^(-2/3)