What is the rate of change of temperature?

Question

Suppose the temperature in degrees Celsius at a point (x, y) is described by a function T(x, y) satisfying Tx(2, 7)=4, Ty(2, 7)=2. The position of a crawling ant after t seconds is given by x(t)=sqrt(1+t), y(t)=-2+3t. After 3 seconds, what is the rate of change of temperature along the ant's path in degrees Celsius per second?

I know that (x(3), y(3))=(2, 7)

Answer 1

∂x/∂t = 1/[2sqrt(1+t)] = 1/4 at t = 3

∂y/∂t = 3

dT/dt = (∂T/∂x)(∂x/∂t) + (∂T/∂y)(∂y/∂t) = (4)(1/4) + 2(3) = 7 degrees of C/sec <==Answer

Answer 2

Rate of change in temperature is defined by derivative

                                       dT/dt = (dT/dx) dx/dt + (dT/dy)dy/dt

where dT/dx = T_x   and dT/dy  =T_y   But 

                                      dx/dt = (1/2)(1/sqrt(1+t)) = 1/4 after 3 seconds

   and

                                                                dy/dt = 3

Thus, we have dT/dt = (1/4)x4 +3x2 = 7