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May 6, 2021

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Section 2-6 : 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