Second order partial derivative and cross second-order partial derivative I'm following a paper (Full text available here!) where at some point (pag.17 and 20) the author get the following derivative:

$$\frac{\partial V}{\partial L}=Y-X\frac{dY}{dX}=\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}$$

where: $Y=\frac{V}{L}$ and $X=\frac{K}{L}$

Then, starting from this he calculates the partial derivative with respect
to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below:

$$\frac{\partial^2 V}{L^{2}}=-\frac{\alpha }{bL} X^{-\frac{c}{b}}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$

$$\frac{\partial^2 V}{dKdL}=\frac{\alpha }{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$

I was stuck trying to derive these latter derivatives. Is there anyone who can help me with this? Thank you so much!
Is there anyone who can help me with this?


Here it is. Equations 1-3, and 5-6 are obtained in preparation for the 2nd derivatives of V with respect to L and K.



Let me know if you have any questions.