We consider the production function $P(L,K)=\dfrac{KL}{4K+L}$ and determine the marginal rate of technical substitution at $(L,K)=(2,3)$.
$P'_L(L,K)=\dfrac{K(4K+L)-KL}{(4K+L)^2}$ and $P'_K(L,K)=\dfrac{L(4K+L)-4KL}{(4K+L)^2}$. Therefore, $MRTS(L,K)=\dfrac{P'_L(L,K)}{P'_K(L,K)}=\dfrac{4K^2}{L^2}$.
Consequently, $MRTS(2,3)=\dfrac{4\cdot 3^2}{2^2}=9$.
This implies that if labor at $(L,K)=(2,3)$ increases by one-tenth of a unit, $\Delta L=\frac{1}{10}$, and the input of capital decreases simultaneously by $\frac{9}{10}$ units, then the output will approximately not change.
$P'_L(L,K)=\dfrac{K(4K+L)-KL}{(4K+L)^2}$ and $P'_K(L,K)=\dfrac{L(4K+L)-4KL}{(4K+L)^2}$. Therefore, $MRTS(L,K)=\dfrac{P'_L(L,K)}{P'_K(L,K)}=\dfrac{4K^2}{L^2}$.
Consequently, $MRTS(2,3)=\dfrac{4\cdot 3^2}{2^2}=9$.
This implies that if labor at $(L,K)=(2,3)$ increases by one-tenth of a unit, $\Delta L=\frac{1}{10}$, and the input of capital decreases simultaneously by $\frac{9}{10}$ units, then the output will approximately not change.