# Math Help - a quick question about partial derivatives

1. ## a quick question about partial derivatives

When you take the partial derivative of

z=x^2y + xy^2 , x=2+t^4 , y= 1-t^3

you get 4(2xy+y^2)t^3 - 3(x^2+2xy)t^2

do you have to plug "2+t^4" in for each x and "1-t^3" in for each y?

Why or why not?

I was under the assumption that you needed to leave the answer in the same variable, in this case "t".

2. In our case, we are calculating not a partial derivative of $z$ but the ordinary derivative of $z$ with respect to $t$. To answer your question, we don't have to replace $x$ and $y$ with expressions for $t$ if we regard them as functions of $t$, as long as we remember that

$\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$

is a function of one variable, $t$.