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. $$f = f\left( {x,y} \right)\hspace{0.5in}x = {u^2} - {v^2},\,\,\,\,\,\,\,y = \frac{u}{v}$$