Section 13.6 : Chain Rule

1. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dz}}{{dt}}\) . \[z = {{\bf{e}}^{{x^{\,2}} - y}}\,\hspace{0.5in}x = \sin \left( {4t} \right),\,\,\,\,y = {t^3} - 9\]
2. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dw}}{{dt}}\) . \[w = {x^4} - 4x{y^2} + {z^3}\,\hspace{0.5in}x = \sqrt t ,\,\,\,\,y = {{\bf{e}}^{2t}},\,\,\,\,z = \frac{1}{t}\]
3. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dw}}{{dt}}\) . \[w = \frac{{4x}}{{y\,{z^3}}}\,\hspace{0.5in}x = 7t - 1,\,\,\,\,y = 1 - 2t,\,\,\,\,z = {t^4}\]
4. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dz}}{{dx}}\) . \[z = 2{x^3}{{\bf{e}}^{4y}}\,\hspace{0.5in}y = \cos \left( {6x} \right)\]
5. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{dz}}{{dx}}\) . \[z = \tan \left( {\frac{x}{y}} \right)\,\hspace{0.5in}y = {{\bf{e}}^{{x^{\,2}}}}\]
6. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{\partial z}}{{\partial u}}\) and \(\displaystyle \frac{{\partial z}}{{\partial v}}\) . \[z = x\sin \left( {{y^2} - x} \right)\,\hspace{0.5in}x = 3u - {v^2},\,\,\,\,y = {u^6}\]
7. Given the following information use the Chain Rule to determine \({w_u}\) and \({w_v}\) . \[w = {x^4}{y^{ - 3}}{z^2}\,\hspace{0.5in}x = {u^2}v,\,\,\,\,y = 3 - uv,\,\,\,\,\,\,z = 7{u^2} - 10v\]
8. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{\partial z}}{{\partial t}}\) and \(\displaystyle \frac{{\partial z}}{{\partial s}}\) . \[z = 6x + {y^2}\tan \left( x \right)\,\hspace{0.5in}x = {p^2} - 3t,\,\,\,\,y = {s^2} - {t^2},\,\,\,\,p = {{\bf{e}}^{3s}}\]
9. Given the following information use the Chain Rule to determine \({w_p}\) and \({w_t}\) . \[w = {x^2}{y^4}{z^6} - 2xy\,\hspace{0.5in}x = 2p,\,\,\,\,y = 3tq,\,\,\,\,\,\,z = 3t{p^2},\,\,\,\,q = 2t\]
10. Given the following information use the Chain Rule to determine \(\displaystyle \frac{{\partial w}}{{\partial u}}\) and \(\displaystyle \frac{{\partial w}}{{\partial v}}\) . \[w = \frac{{\sqrt y }}{{{x^2}{z^3}}}\,\hspace{0.5in}x = uv,\,\,\,\,y = {u^2} - {p^3},\,\,\,\,\,\,z = 4qp,\,\,\,\,p = 2u - 3v,\,\,\,\,\,\,q = {v^2}\]
11. Determine formulas for \({w_u}\) and \({w_t}\) for the following situation. \[w = w\left( {x,y} \right)\hspace{0.5in}x = x\left( {y,z} \right),\,\,\,\,y = y\left( {u,v} \right),\,\,\,\,z = z\left( {u,t} \right),\,\,\,\,v = v\left( t \right)\]
12. Determine formulas for \(\displaystyle \frac{{\partial w}}{{\partial s}}\) and \(\displaystyle \frac{{\partial w}}{{\partial t}}\) for the following situation. \[w = w\left( {x,y,z} \right)\hspace{0.5in}x = x\left( {u,v,t} \right),\,\,\,\,y = y\left( p \right),\,\,\,\,z = z\left( {u,t} \right),\,\,\,\,v = v\left( {p,t} \right),\,\,\,\,p = p\left( {s,t} \right)\]
13. Compute \(\displaystyle \frac{{dy}}{{dx}}\) for the following equation. \[\cos \left( {2x + 3y} \right) = {x^5} - 8{y^2}\]
14. Compute \(\displaystyle \frac{{dy}}{{dx}}\) for the following equation. \[\cos \left( {2x} \right)\sin \left( {3y} \right) - xy = {y^4} + 9\]
15. Compute \(\displaystyle \frac{{\partial z}}{{\partial x}}\) and \(\displaystyle \frac{{\partial z}}{{\partial y}}\) for the following equation. \[{z^3}{y^4} - {x^2}\cos \left( {2y - 4z} \right) = 4z\]
16. Compute \(\displaystyle \frac{{\partial z}}{{\partial x}}\) and \(\displaystyle \frac{{\partial z}}{{\partial y}}\) for the following equation. \[\sin \left( x \right){{\bf{e}}^{4x\,z}} + 2{z^2}y = \cos \left( z \right)\]
17. Determine \({f_{u\,u}}\) and \({f_{v\,v}}\) for the following situation. \[f = f\left( {x,y} \right)\hspace{0.5in}x = {{\bf{e}}^u}sin\left( v \right),\,\,\,\,\,\,\,y = {{\bf{e}}^u}\cos \left( v \right)\]
18. Determine \({f_{u\,u}}\) and \({f_{v\,v}}\) for the following situation. \[f = f\left( {x,y} \right)\hspace{0.5in}x = {u^2} - {v^2},\,\,\,\,\,\,\,y = \frac{u}{v}\]