91Ó°ÊÓ

If \(z=5 x^{4}+2 x^{3} y^{2}-3 y\) find (a) \(\frac{\partial z}{\partial x}\) and (b) \(\frac{\partial z}{\partial y}\)

Given \(y=4 \sin 3 x \cos 2 t\), find \(\frac{\partial y}{\partial x}\) and \(\frac{\partial y}{\partial t}\)

If \(z=\sin x y\) show that $$ \frac{1}{y} \frac{\partial z}{\partial x}=\frac{1}{x} \frac{\partial z}{\partial y} $$

Determine \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) when $$ z=\frac{1}{\sqrt{\left(x^{2}+y^{2}\right)}} $$

Pressure \(p\) of a mass of gas is given by \(p V=m R T\), where \(m\) and \(R\) are constants, \(V\) is the volume and \(T\) the temperature. Find expressions for \(\frac{\partial p}{\partial T}\) and \(\frac{\partial p}{\partial V}\)

Given \(z=4 x^{2} y^{3}-2 x^{3}+7 y^{2}\) find (a) \(\frac{\partial^{2} z}{\partial x^{2}}\) (b) \(\frac{\partial^{2} z}{\partial y^{2}}\) (c) \(\frac{\partial^{2} z}{\partial x \partial y}\) (d) \(\frac{\partial^{2} z}{\partial y \partial x}\)

Show that when \(z=\mathrm{e}^{-t} \sin \theta\) (a) \(\frac{\partial^{2} z}{\partial t^{2}}=-\frac{\partial^{2} z}{\partial \theta^{2}}\), and (b) \(\frac{\partial^{2} z}{\partial t \partial \theta}=\frac{\partial^{2} z}{\partial \theta \partial t}\)

Show that if \(z=\frac{x}{y} \ln y\), then (a) \(\frac{\partial z}{\partial y}=x \frac{\partial^{2} z}{\partial y \partial x}\) and \((\mathrm{b})\) evaluate \(\frac{\partial^{2} z}{\partial y^{2}}\) when \(x=-3\) and \(y=1\)