Use chain rule of differentiation and find derivative of f(x)=(x^3+4)^4?

To use the chain rule, we'll specify first that f(x) is
the result of composition of 2 functions.

u(x) = x^3 + 4
and v(u) = u^4

f(x) = (vou)(x) = v(u(x)) = v(x^3 + 4) =
(x^3 + 4)^4

We'll differentiate f(x) and we'll
get:

f'(x) =
v'(u(x))*u'(x)

First, we'll differentiate v with respect to
u:

v'(u) = 4u^(4-1) =
4u^3

Second, we'll differentiate u with respect to
x:

u'(x) = (x^3 + 4)' =
3x^2

f'(x) = 4u^3*3x^2

We'll
substitute u and we'll get:

The derivative of
f(x) is: f'(x) = 12x^2*( x^3 + 4)^3.