# a quick question about partial derivatives

• Oct 28th 2009, 06:24 PM
yoleven
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".
• Oct 28th 2009, 06:47 PM
Scott H
In our case, we are calculating not a partial derivative of $\displaystyle z$ but the ordinary derivative of $\displaystyle z$ with respect to $\displaystyle t$. To answer your question, we don't have to replace $\displaystyle x$ and $\displaystyle y$ with expressions for $\displaystyle t$ if we regard them as functions of $\displaystyle t$, as long as we remember that

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

is a function of one variable, $\displaystyle t$.