Answer
$g'(x)=\frac{27x^2-3}{9x^2+1}-\frac{8x^2-2}{4x^2+1}$
Work Step by Step
Hint: $\frac{d}{dx}(\int_{a(x)}^{b(x)}f(u)du)=\frac{d}{dx}(F(b(x))-F(a(x))=f(b(x))\cdot b'(x)-f(a(x))\cdot a'(x)$
Given: $f(u)=\frac{u^2-1}{u^2+1}$, $a(x)=2x$, and $b(x)=3x$
Find $f(a(x))$:
$f(2x)=\frac{(2x)^2-1}{(2x)^2+1}=\frac{4x^2-1}{4x^2+1}$
Find $f(b(x))$:
$f(3x)=\frac{(3x)^2-1}{(3x)^2+1}=\frac{9x^2-1}{9x^2+1}$
Find $a'(x)$ and $b'(x)$:
$a'(x)=\frac{d}{dx}(2x)=2$
$b'(x)=\frac{d}{dx}(3x)=3$
Find $g'(x)$:
$g'(x)=f(3x)\cdot b'(x)-f(2x)\cdot a'(x)$
$g'(x)=\frac{9x^2-1}{9x^2+1}\cdot 3-\frac{4x^2-1}{4x^2+1}\cdot 2$
$g'(x)=\frac{27x^2-3}{9x^2+1}-\frac{8x^2-2}{4x^2+1}$
