Example 9 Let $f(x): \mathbf{R}^{+} \rightarrow \mathbf{R}^{+}$, solve the functional inequality $f^{\prime}(x)(x-1)>0$.


Solve the function $p(x)\left(\forall p(x)>0, x \in \mathbf{R}^{+}, x \neq 1\right)$, let $f^{\prime}(x)(x-1)=p(x)$, then $f^{\prime}(x)=$ $\frac{p(x)}{x-1}$, so the general solution of the functional inequality is $f(x)=\int \frac{p(x)}{x-1} \mathrm{~d} x \quad(p(x)>0, \forall x \in \mathbf{R}, x \neq 1)$.