I need help with the following problem: the temperature in a given point in $\mathbb{R}^3$ is given by $T(x,y,z) = x^2+y^2+z^2$. And a particle's position at a point in time $t$ is given parametrically by $(x,y,z) = (t, t^2, t^3)$.
I'm given the task of finding the rate of change in temperature when $t = \frac{1}{2}$. Since the rate of change is given by the derivative, my guess would be to find the partial derivatives for $x$, $y$, and for $z$ and later substitute the coordinates $(\frac{1}{2},\frac{1}{4}, \frac{1}{8})$. But I wouldn't know exactly how. Any help is most welcome.
$\textbf{Hint}$: Use Chain Rule.
\begin{align*} \frac{d g}{dt} = d_{c(t)}T \cdot d_t c = \begin{pmatrix}\dfrac{\partial T}{\partial x} &\dfrac{\partial T}{\partial y}&\dfrac{\partial T}{\partial z}\end{pmatrix}_{c(t)}\ \begin{pmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \\ \frac{dz}{dt}\end{pmatrix}\end{align*}