If $w=x^{2}y+y^{2}z,$ where $x=t$, $y=t^{2}$, and $z=t^{3}$, then $w$ is a function of $t,$ and $$\begin{eqnarray*} \dfrac{dw}{dt}& =&\dfrac{\partial w}{\partial x}\dfrac{dx}{dt}+\dfrac{\partial w}{ \partial y}\dfrac{dy}{dt}+\dfrac{\partial w}{\partial z}\dfrac{dz}{dt}=\, \underset{\color{#0066A7}{\dfrac{\partial w}{\partial x}}}{\underbrace{(2xy)}}\underset{\color{#0066A7}{\dfrac{ dx}{dt}}}{\underbrace{(1)}}+\underset{\color{#0066A7}{\dfrac{\partial w}{\partial y}}}{ \underbrace{(x^{2}+2yz)}}\underset{\color{#0066A7}{\dfrac{dy}{dt}}}{\underbrace{(2t)}}+ \underset{\color{#0066A7}{\dfrac{\partial w}{\partial z}}}{\underbrace{(y^{2})}}\underset{\color{#0066A7}{ \dfrac{dz}{dt}}}{\underbrace{(3t^{2})}} \nonumber \\ &=& 2t^{3}+(t^{2}+2t^{5})(2t)+(t^{4})(3t^{2})=7t^{6}+4t^{3} \nonumber \end{eqnarray*}$$