First mean value theorem for definite integrals and differentiation

Question

From the First MVT for integral we have ($g$ with constant sign) $\int_{a}^{b}f(x)g(x)\,dx=f(c)\int _{a}^{b}g(x)\,dx.$. My question: is there a theorem that tells how to sign $\frac{dc}{db}$. Is the below differentiation, using Leibniz rule, correct ? \begin{eqnarray*} \frac{d}{db}\left( \int_{a}^{b}f(x)g(x)\,dx\right) &=&f\left( b\right) g\left( b\right) \\ &=&\frac{dc}{db}f^{\prime }(c)\int_{a}^{b}g(x)\,dx+f(c)g(b) \end{eqnarray*} So \begin{equation*} \frac{dc}{db}=\frac{\left[ f\left( b\right) -f(c)\right] g(b)}{f^{\prime }(c)\int_{a}^{b}g(x)\,dx} \end{equation*}