The fractional derivative of $x^k$, for $k>0$, is
$$\frac{d^a}{dx^a}x^k = \frac{\Gamma{(k+1)}}{\Gamma{(k-a+1)}}x^{k-a}$$
Variabilising the order of differentiation by making it a function of $x$, we can write equations of which each solution, if any solutions exist, send one function to another by differentiating it, giving
$$\frac{d ^{u(x)}}{dx^{u(x)}} f(x) = g(x)$$
An example of a specific case is
$$\frac{d ^{u(x)}}{dx^{u(x)}} x^2 = x.$$
Putting $k=2$ into the definition of the fractional derivative, we get
$$\frac{d^a}{dx^a}x^2 = \frac{\Gamma{(3)}}{\Gamma{(3-a)}}x^{2-a}=\frac{2x^{2-a}}{\Gamma{(3-a)}}.$$
Writing $a$ as $u(x)$, putting the expression as equal to $x$, and assuming the denominator is non-zero, we get:
$$2x^{1-u(x)}=\Gamma{(3-u(x))}.$$
Writing $3-u(x)$ as $v$, we get
$$2x^{v-2}=\Gamma(v).$$
When $x=1$, one solution is $u=3$ and another is $u=0.443$;
when $x=2$, we get $u=1$ and $u=4.46$;
when $x=2.1$, $u=1.04$ and $u=4.64$.
What can we say about $u(x)$? What can we say more generally about $u(x)$, for example if rather than sending $x^2$ to $x$, we send $\sin(x)$ to $x$?