Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for $L_p\left(T\right)$ and $L_p[-1,1]$ for 0 < p < ∞ using the moduli of smoothness ${\omega }^r{(f,t)}_p$ and ${\omega }_{\phi }^r{(f,t)}_p$ respectively.