1. Since $g$ is the inverse of $f$, we use the inverse-function derivative formula:

$g'(x)=\frac{1}{f'(g(x))}$

This follows from

$f(g(x))=x$

Differentiating both sides:

$f'(g(x))\cdot g'(x)=1$

Hence,

$g'(x)=\frac{1}{f'(g(x))}.$

2. Given

$f'(x)=\frac{1}{1+x^5},$

replace $x$ by $g(x)$:

$f'(g(x))=\frac{1}{1+\{g(x)\}^5}.$

3. Therefore,

$g'(x)=\frac{1}{\dfrac{1}{1+\{g(x)\}^5}}=1+\{g(x)\}^5.$

4. Compare with the options:

• A: $\dfrac{1}{1+\{g(x)\}^5}$
• B: $1+\{g(x)\}^5$
• C: $1+x^5$
• D: $5x^4$

So the correct option is:

$\boxed{\text{B}}$