The temperature at a point $(x,y)$ on a metal plate in the $xy$-plane is $T(x,y)=x^3+2y^2+x$. Assume that distance is measured in centimeters and find the rate at which temperatures changes with respect to distance if we start at the point $P(1,2)$ and move
a) to the right and parallel to the x-axis b) upward and parallel to the y-axis
My Attempt
Part a) I wonder if I got the parts mixed up \begin{align}T_x(x,y)&=3x^2+1 \\ T_x(1,2)&=3+1=4 \end{align} Part b) \begin{align} T_y(x,y)&=4y^2 \\ T_y(1,2)&=4(2)=8 \end{align}
$a)$ You first need to find the unit vector of this direction. Take $Q = (2,2)$, then $u = \vec{PQ}= <1,0>$, and the rate of change you are looking at is $1\cdot f_x= 3x^2+1|_{x=1}= 4$. ( The $1$ in front comes from the dot product of the gradient of $f$ with $u$).
$b)$ Similarly you have the rate of change is $1\cdot f_y= 4y|_{y=2} = 8$.