Answer
$$\frac{dw}{dt}=26t$$
When $t=2$
$$\frac{dw}{dt}=52$$
Work Step by Step
Using the chain rule we have
$$\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}=\frac{\partial }{\partial x}(x^2+y^2)\frac{d}{dt}(2t)+\frac{\partial }{\partial y}(x^2+y^2)\frac{d}{dt}(3t)=2x\cdot2+2y\cdot3=4x+6y$$
Expressing this in terms of $t$ we have:
$$\frac{dw}{dt}=4\cdot2t+6\cdot3t=8t+18t=26t$$
When $t=2$ we have:
$$\frac{dw}{dt}=26\cdot2=52$$
