Question

if u(t,v(t,p)) , what is the partial derivative of u with respect to t at constant p?

Answer 1

can you really predict this ? it'll vary with every other function,,i guess..

Answer 2

we have to use chain rule here.which is confusing me

Answer 3

Here is a picture of the function and it's arguments in tree-form. From this you can deduce all partial derivatives, to get the partial derivative with respect to t, find all t's in the tree and multiply each derivative along the way.

For this we get
\[ \frac{du}{dt}=\left(\frac{\partial u}{\partial t}\right)_p+\frac{\partial u}{\partial v}\left(\frac{\partial v}{\partial t}\right)_p \]
Now you can just solve for \(\left(\frac{\partial u}{\partial t}\right)_p\).