The Chain Rule:

 1. If \( z=x^2y+3xy^3\), where \(x = \sin (t)\) and \(y=\cos (2t)\), find \(\displaystyle{\frac{dz}{dt}}\) when \(t=0\).   solution

2. The pressure \( P \)  of a gas is related to the volume \(V\)  and temperature \(T\)  by the formula \( PV = kT\), where temperature is expressed in kelvins \(K\).  Find \(\displaystyle{\frac{dP}{dt}}\) when \(k = 1, \frac{dV}{dt}  = 3\, cm^3 / min, \, \frac{dT}{dt} =0.8 K/min, V = 20 cm^3 \, \)  and  \( \: T = 300^\circ K\).   solution

3. If \( z=2\sin(xy)+xy^3\), where \(x = 2u-3v\) and \(y=1+e^{uv}\), find \(\displaystyle{\frac{\partial z}{\partial u}}\) and \(\displaystyle{\frac{\partial z}{\partial v}}\).    solution

4. Let \(\displaystyle{z=e^{x^3y^2}} \), where \(x = u+v\) and \(y=\sqrt{uv}\), find \(\displaystyle{\frac{\partial z}{\partial u}}\) and \(\displaystyle{\frac{\partial z}{\partial v}}\).   solution

5.  Find \(\displaystyle{\frac{dy}{dx}}\) using partial derivatives for the following. \( \quad \displaystyle{ \sin(xy^2)+1=2y^3-3x^2}\)  solution

6. Calculate \(\displaystyle{\frac{\partial z}{\partial x}}\) and \(\displaystyle{\frac{\partial z}{\partial y}}\) for the surface \( z = \dfrac{x^2e^y}{ye^x}\).   solution