If u=(fog)(x) and v=(gof)(x), verify if u'=v'? f(x)=3x+2,g(x)=x^2+1

The answer is very simple: the derivatives of the
functions u and v are not equal.

Since we know that the
composition of 2 functions is not commutative, then the derivatives of 2 expressions
that are not matching, are not equal.

We'll see how it
happens.

We'll compose (fog)(x) =
f(g(x))

We'll substitute x by g(x), in the expression of
f(x).

f(g(x)) = 3(x^2 + 1) +
2

We'll remove the
brackets:

f(g(x)) = 3x^2 +
5

u(x) = f(g(x))

We'll
differentiate with respect to x:

u'(x) =
6x

Now, we'll compose (gof)(x) =
g(f(x))

We'll substitute x by f(x), in the expression of
g(x).

g(f(x)) = (3x + 2)^2 +
1

We'll expand the
square:

g(f(x)) = 9x^2 + 12x + 4 +
1

g(f(x)) = 9x^2 + 12x +
5

v(x) = g(f(x))

We'll
differentiate with respect to x:

v'(x) = 18x +
12

We can see that the expression of u'(x) =
6x and the expression of v'(x) = 18x + 12 are not
equal.