Is there a solution to $(f\circ g)'(x)=(f'\circ g')(x)$?

Question

So the chain rule is defined as $$ \frac{df(g(x))}{dx}=\frac{df}{dg}\frac{dg}{dx} $$ Is there a function where the derivative of a function composition is the composition of the derivatives, essentially solving this DE: $$ \frac{df}{dg}\frac{dg}{dx}=\frac{df(\frac{dg}{dx})}{dx}=(f'\circ g')(x)=f'(g'(x)) $$