Let the temperature of a point in the $xy$-plane be given by $T(x,y)=x^2+y^2$. What is the speed of temperature change in point $(1,5)$ in direction that makes a $30^\circ$ angle with the positive part of the $x$-axis ?
I have tried using vector calculus, but didn't make any result at the end? What is the best way to do this?
$$\Delta T=\Delta x\frac{\partial T}{\partial x}|_{1,5}+\Delta y\frac{\partial T}{\partial y}|_{1,5}$$ $$\frac{\Delta T}{\Delta s}=\frac{\Delta x}{\Delta s} \frac{\partial T}{\partial x}|_{1,5}+\frac{\Delta y}{\Delta s} \frac{\partial T}{\partial y}|_{1,5}=\cos\theta\frac{\partial T}{\partial x}|_{1,5}+\sin\theta\frac{\partial T}{\partial y}|_{1,5}$$