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