Answer
$\dfrac{dL}{dt}=\dfrac{31}{13}$
Work Step by Step
Let us differentiating $L$ with respect to t, then
$\dfrac{dL}{dt}=\dfrac{1}{2}\dfrac{1}{\sqrt{(x^2+y^2)}}(2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt})$
Since,$L=\sqrt{x^2+y^2}$,$\dfrac{dx}{dt}=-1$and $\dfrac{dy}{dt}=3$
so, $\dfrac{dL}{dt}=\dfrac{1}{\sqrt{(x^2+y^2)}}[(x(-1)+y(3))]$
Plug $x=5$ and $y=12$
Thus, $\dfrac{dL}{dt}=\dfrac{1}{\sqrt{(5^2+12^2)}}(5(-1)+12(3))=\frac{31}{\sqrt{169}}=\dfrac{31}{13}$
