Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dz}}{{dt}}\) .
\[z = \cos \left( {y\,{x^2}} \right)\,\hspace{0.5in}x = {t^4} - 2t,\,\,\,\,y = 1 - {t^6}\]
Solution
Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dw}}{{dt}}\) .
\[w = \frac{{{x^2} - z}}{{{y^4}}}\,\hspace{0.5in}x = {t^3} + 7,\,\,\,\,y = \cos \left( {2t} \right),\,\,\,\,z = 4t\]
Solution
Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dz}}{{dx}}\) .
\[z = {x^2}{y^4} - 2y\,\hspace{0.5in}y = \sin \left( {{x^2}} \right)\]
Solution
Given the following information use the Chain Rule to determine \(\displaystyle \frac{{\partial z}}{{\partial u}}\) and \(\displaystyle \frac{{\partial z}}{{\partial v}}\) .
\[z = {x^{ - 2}}{y^6} - 4x\,\hspace{0.5in}x = {u^2}v,\,\,\,\,y = v - 3u\]
Solution
Given the following information use the Chain Rule to determine \({z_t}\) and \({z_p}\) .
\[z = 4y\sin \left( {2x} \right)\,\hspace{0.5in}x = 3u - p,\,\,\,\,y = {p^2}u,\,\,\,\,\,\,u = {t^2} + 1\]
Solution
Given the following information use the Chain Rule to determine \(\displaystyle \frac{{\partial w}}{{\partial t}}\) and \(\displaystyle \frac{{\partial w}}{{\partial s}}\) .
\[w = \sqrt {{x^2} + {y^2}} + \frac{{6z}}{y}\,\hspace{0.5in}x = \sin \left( p \right),\,\,\,\,y = p + 3t - 4s,\,\,\,\,z = \frac{{{t^3}}}{{{s^2}}},\,\,\,\,p = 1 - 2t\]
Solution
Determine formulas for \(\displaystyle \frac{{\partial w}}{{\partial t}}\) and \(\displaystyle \frac{{\partial w}}{{\partial v}}\) for the following situation.
\[w = w\left( {x,y} \right)\hspace{0.5in}x = x\left( {p,q,s} \right),\,\,\,\,y = y\left( {p,u,v} \right),\,\,\,\,s = s\left( {u,v} \right),\,\,\,\,p = p\left( t \right)\]
Solution
Determine formulas for \(\displaystyle \frac{{\partial w}}{{\partial t}}\) and \(\displaystyle \frac{{\partial w}}{{\partial u}}\) for the following situation.
\[w = w\left( {x,y,z} \right)\hspace{0.5in}x = x\left( t \right),\,\,\,\,y = y\left( {u,v,p} \right),\,\,\,\,z = z\left( {v,p} \right),\,\,\,\,v = v\left( {r,u} \right),\,\,\,\,p = p\left( {t,u} \right)\]
Solution
Compute \(\displaystyle \frac{{dy}}{{dx}}\) for the following equation.
\[{x^2}{y^4} - 3 = \sin \left( {xy} \right)\]
Solution
Compute \(\displaystyle \frac{{\partial z}}{{\partial x}}\) and \(\displaystyle \frac{{\partial z}}{{\partial y}}\) for the following equation.
\[{{\bf{e}}^{z\,y}} + x{z^2} = 6x{y^4}{z^3}\]
Solution
Determine \({f_{u\,u}}\) for the following situation.
\[f = f\left( {x,y} \right)\hspace{0.5in}x = {u^2} + 3v,\,\,\,\,\,\,\,y = uv\]
Solution