Section 13.6 : Chain Rule

1. 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
2. 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
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
9. Compute $\displaystyle \frac{{dy}}{{dx}}$ for the following equation. $${x^2}{y^4} - 3 = \sin \left( {xy} \right)$$ Solution
10. 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
11. Determine ${f_{u\,u}}$ for the following situation. $$f = f\left( {x,y} \right)\hspace{0.5in}x = {u^2} + 3v,\,\,\,\,\,\,\,y = uv$$ Solution