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 13.6 : Chain Rule
10. Compute \(\displaystyle \frac{{\partial z}}{{\partial x}}\) and \(\displaystyle \frac{{\partial z}}{{\partial y}}\) for the following equation.
\[{{\bf{e}}^{z\,y}} + x{z^2} = 6x{y^4}{z^3}\]Show All Steps Hide All Steps
Start SolutionFirst a quick rewrite of the equation.
\[{{\bf{e}}^{z\,y}} + x{z^2} - 6x{y^4}{z^3} = 0\] Show Step 2From the rewrite in the previous step we can see that,
\[F\left( {x,y} \right) = {{\bf{e}}^{z\,y}} + x{z^2} - 6x{y^4}{z^3}\]We can now simply use the formulas we derived in the notes to get the derivatives.
\[\frac{{\partial z}}{{\partial x}} = - \frac{{{F_x}}}{{{F_z}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{{z^2} - 6{y^4}{z^3}}}{{y{{\bf{e}}^{z\,y}} + 2xz - 18x{y^4}{z^2}}} = \frac{{6{y^4}{z^3} - {z^2}}}{{y{{\bf{e}}^{z\,y}} + 2xz - 18x{y^4}{z^2}}}}}\] \[\frac{{\partial z}}{{\partial y}} = - \frac{{{F_y}}}{{{F_z}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - \frac{{z{{\bf{e}}^{z\,y}} - 24x{y^3}{z^3}}}{{y{{\bf{e}}^{z\,y}} + 2xz - 18x{y^4}{z^2}}} = \frac{{24x{y^3}{z^3} - z{{\bf{e}}^{z\,y}}}}{{y{{\bf{e}}^{z\,y}} + 2xz - 18x{y^4}{z^2}}}}}\]Note that in for the second form of each of the answers we simply moved the “-” in front of the fraction up to the numerator and multiplied it through. We could just have easily done this with the denominator instead if we’d wanted to.