I know that, according to the statement of the Leibniz rule for the derivation of a parametric integral, if we have:
$$F(\lambda)=\int^{b(\lambda)}_{a(\lambda)}f(\lambda,x)dx$$
then the following equality holds (I omit the hypotheses of the theorem for the sake of brevity):
$$\dfrac{\partial F}{\partial\lambda_k}=\int^{b(\lambda)}_{a(\lambda)}\dfrac{\partial f}{\partial\lambda_k}dx+\dfrac{\partial b}{\partial\lambda_k}\cdot f(\lambda,b(\lambda))-\dfrac{\partial a}{\partial\lambda_k}\cdot f(\lambda,a(\lambda))$$
I want to apply this rule in order to calculate the following integral that arises in the context of a physics problem:
$$\int^{a(\tau)}_0\dfrac{1}{g^3}da'$$
where:
$$g(a)=\dfrac{1}{a}\dfrac{da}{d\tau}$$
So, since we don't have an independent integration variable and parameter here, but rather an integration variable $a(\tau)$ which depends on the parameter $\tau$, which is the correct way to apply Leibniz rule here? My attempt is:
$$\dfrac{\partial}{\partial\tau}\bigg(\int^a_0\dfrac{da'}{g^3}\bigg)=\int^a_0\dfrac{\partial}{\partial\tau}\bigg(\dfrac{1}{g^3}\bigg)da'+\dfrac{da}{d\tau}\cdot\dfrac{1}{g^3}$$
Am I right? Help, please? Thank you very much in advance!