Question

use the chain rule to find the indicated partial derivatives.

$p = \sqrt { u ^ { 2 } + v ^ { 2 } + w ^ { 2 } }$, $u = x e ^ { y }$, $v = y e ^ { x }$, $w = e ^ { x y }$;

$\frac { \partial p } { \partial x }$, $\frac { \partial p } { \partial y }$ when $x = 0$, $y = 3$

$\frac { \partial p } { \partial x } =$

$\frac { \partial p } { \partial y } =$

Explanation:

Step1: Calculate the partial derivatives of \(u\), \(v\), \(w\)

• For \(u = xe^{y}\), \(\frac{\partial u}{\partial x}=e^{y}\), \(\frac{\partial u}{\partial y}=xe^{y}\)
• For \(v = ye^{x}\), \(\frac{\partial v}{\partial x}=ye^{x}\), \(\frac{\partial v}{\partial y}=e^{x}\)
• For \(w = e^{xy}\), \(\frac{\partial w}{\partial x}=ye^{xy}\), \(\frac{\partial w}{\partial y}=xe^{xy}\)

Step2: Calculate \(\frac{\partial p}{\partial u}\), \(\frac{\partial p}{\partial v}\), \(\frac{\partial p}{\partial w}\)

Since \(p=\sqrt{u^{2}+v^{2}+w^{2}}=(u^{2}+v^{2}+w^{2})^{\frac{1}{2}}\)
\(\frac{\partial p}{\partial u}=\frac{u}{\sqrt{u^{2}+v^{2}+w^{2}}}\), \(\frac{\partial p}{\partial v}=\frac{v}{\sqrt{u^{2}+v^{2}+w^{2}}}\), \(\frac{\partial p}{\partial w}=\frac{w}{\sqrt{u^{2}+v^{2}+w^{2}}}\)

Step3: Substitute \(x = 0\), \(y = 3\)

• \(u=xe^{y}=0\), \(v = ye^{x}=3\), \(w = e^{xy}=1\)
• \(\sqrt{u^{2}+v^{2}+w^{2}}=\sqrt{0 + 9+1}=\sqrt{10}\)
• \(\frac{\partial p}{\partial u}=0\), \(\frac{\partial p}{\partial v}=\frac{3}{\sqrt{10}}\), \(\frac{\partial p}{\partial w}=\frac{1}{\sqrt{10}}\)

Step4: Use the chain - rule \(\frac{\partial p}{\partial x}=\frac{\partial p}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial p}{\partial v}\frac{\partial v}{\partial x}+\frac{\partial p}{\partial w}\frac{\partial w}{\partial x}\)

Substitute the values:
\(\frac{\partial p}{\partial x}=0\times e^{y}+\frac{3}{\sqrt{10}}\times ye^{x}+\frac{1}{\sqrt{10}}\times ye^{xy}\)
When \(x = 0\), \(y = 3\)
\(\frac{\partial p}{\partial x}=0+\frac{3}{\sqrt{10}}\times3\times1+\frac{1}{\sqrt{10}}\times3\times1=\frac{9 + 3}{\sqrt{10}}=\frac{12}{\sqrt{10}}=\frac{6\sqrt{10}}{5}\)

Step5: Use the chain - rule \(\frac{\partial p}{\partial y}=\frac{\partial p}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial p}{\partial v}\frac{\partial v}{\partial y}+\frac{\partial p}{\partial w}\frac{\partial w}{\partial y}\)

Substitute the values:
\(\frac{\partial p}{\partial y}=0\times xe^{y}+\frac{3}{\sqrt{10}}\times e^{x}+\frac{1}{\sqrt{10}}\times xe^{xy}\)
When \(x = 0\), \(y = 3\)
\(\frac{\partial p}{\partial y}=0+\frac{3}{\sqrt{10}}\times1+0=\frac{3\sqrt{10}}{10}\)

Answer:

\(\frac{\partial p}{\partial x}=\frac{6\sqrt{10}}{5}\)
\(\frac{\partial p}{\partial y}=\frac{3\sqrt{10}}{10}\)