This paper presents a new lemma for Caputo fractional derivatives when \(0 < \alpha < 1\). This lemma is useful for applying the fractional-order extension of the Lyapunov direct method to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying. The lemma states that for any continuous and derivable function \(x(t)\), the following inequality holds: \[ \frac{1}{2} \frac{C}{t_0} D_t^\alpha x^2(t) \leq x(t) \frac{C}{t_0} D_t^\alpha x(t), \quad \forall \alpha \in (0, 1) \] The proof of this lemma involves integrating by parts and applying the Leibniz rule for fractional differentiation. The lemma is then used to derive a simple Lyapunov candidate function for proving the stability of fractional order systems. Two examples are provided to illustrate the effectiveness of the lemma in stability analysis. The first example is a fractional order linear time-varying system, and the second example is a fractional order nonlinear system. Both examples demonstrate that the stability of the systems can be proven using the lemma, which is more straightforward than using the classical Lyapunov direct method or other existing techniques. The paper concludes by highlighting the importance of the lemma in finding Lyapunov functions and proving the stability of fractional order systems.This paper presents a new lemma for Caputo fractional derivatives when \(0 < \alpha < 1\). This lemma is useful for applying the fractional-order extension of the Lyapunov direct method to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying. The lemma states that for any continuous and derivable function \(x(t)\), the following inequality holds: \[ \frac{1}{2} \frac{C}{t_0} D_t^\alpha x^2(t) \leq x(t) \frac{C}{t_0} D_t^\alpha x(t), \quad \forall \alpha \in (0, 1) \] The proof of this lemma involves integrating by parts and applying the Leibniz rule for fractional differentiation. The lemma is then used to derive a simple Lyapunov candidate function for proving the stability of fractional order systems. Two examples are provided to illustrate the effectiveness of the lemma in stability analysis. The first example is a fractional order linear time-varying system, and the second example is a fractional order nonlinear system. Both examples demonstrate that the stability of the systems can be proven using the lemma, which is more straightforward than using the classical Lyapunov direct method or other existing techniques. The paper concludes by highlighting the importance of the lemma in finding Lyapunov functions and proving the stability of fractional order systems.