    \begin{align*}
        \frac{\partial c}{\partial \log c_\ell} 
        &= \frac{\frac{\partial c}{\partial \log c_\ell}}{\frac{\partial \log c_\ell}{\partial c_\ell}}\\
        &= c_\ell \cdot \frac{\partial c}{\partial c_\ell}.
    \end{align*}
