Question on applying L'Hopital's rule--Calculus I In the below picture, how does the equality sign work out?
It seems that using L'Hopital's rule, the x on the bottom turns to 1, and the derivative of the expression on the numerator side is equal to the expression on the right-handed side, but I don't see how..
This is from the article: https://www.mdpi.com/2073-8994/10/9/380
Thank you.


$$\begin{align}\lim_{x\to0^+}\frac{\ln\sum_{k=1}^n\lambda_kc_k^x}{x}
&=\lim_{x\to0^+}\frac{\left(\ln\sum_{k=1}^n\lambda_kc_k^x\right)'_x}{(x)'_x}\\
&=\lim_{x\to0^+}\frac{\sum_{k=1}^n\lambda_kc_k^x\ln c_k}{\sum_{k=1}^n\lambda_kc_k^x}\\
&=\frac{\sum_{k=1}^n\lambda_k\ln c_k}{\underbrace{\sum_{k=1}^n\lambda_k}_{=1}}.\\
\end{align}$$
