Answer
$2^{\circ} C \ per \ second$
Work Step by Step
We will re-write the given equation as:$\dfrac{dT(t)}{dt}=T_t[x(t) , y(t)]=T_x[x(t), y(t)] \cdot x'(t)+T_y[x(t), y(t)] \cdot y'(t) \\=T_x[x(t), y(t)] \cdot \dfrac{1}{2\sqrt {1+t}}+ (\dfrac{1}{3}) T_y[x(t), y(t)] $
So, the required rate of the change in temperature can be found as:
$\dfrac{dT(3)}{dt}=T_x(2,3)(\dfrac{1}{4})+T_y(2,3) (\dfrac{1}{3})=(4)(\dfrac{1}{4})+(3) (\dfrac{1}{3}) \\ =2^{\circ} C \ per \ second$
