d d x a = 0 {\displaystyle \frac{d}{dx}a=0} , for some constant a.
The limit definition of the derivative, f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h {\displaystyle f'(x)=\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}}
Let f(x) = a for some constant a. From the limit definition of the derivative:
QED