How to differentiate y=(2+4x^2)^3?

Since the given function is composed, we'll apply chain
rule to differentiate it.

We'll differentiate with respect
to x. First, we'll identify the composed functions, whose final result is
y.

dy/dx =
(dy/dt)*(dt/dx)

We'll put 2+4x^2 =
t

y = t^3

We'll differentiate
with respect to t:

dy/dt =
d(t^3)/dt

dy/dt = 3t^2

We'll
differentiate t with respect to x.

dt/dx =
d(2+4x^2)/dx

dt/dx = 8x

dy/dx
= 3t^2*8x = 24x*t^2

We'll substitute back
t:

dy/dx = 24x(2 +
4x^2)^2