Paul's Online Notes
Home / Calculus III / Partial Derivatives / Chain Rule
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### 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