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Section 13.4 : Higher Order Partial Derivatives

For problems 1 & 2 verify Clairaut’s Theorem for the given function.

1. \(\displaystyle f\left( {x,y} \right) = {x^3}{y^2} - \frac{{4{y^6}}}{{{x^3}}}\) Solution
2. \(\displaystyle A\left( {x,y} \right) = \cos \left( {\frac{x}{y}} \right) - {x^7}{y^4} + {y^{10}}\) Solution

For problems 3 – 6 find all 2nd order derivatives for the given function.

1. \(g\left( {u,v} \right) = {u^3}{v^4} - 2u\sqrt {{v^3}} + {u^6} - \sin \left( {3v} \right)\) Solution
2. \(f\left( {s,t} \right) = {s^2}t + \ln \left( {{t^2} - s} \right)\) Solution
3. \(\displaystyle h\left( {x,y} \right) = {{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} - \frac{{{y^3}}}{x}\) Solution
4. \(\displaystyle f\left( {x,y,z} \right) = \frac{{{x^2}{y^6}}}{{{z^3}}} - 2{x^6}z + 8{y^{ - 3}}{x^4} + 4{z^2}\) Solution
5. Given \(f\left( {x,y,z} \right) = {x^4}{y^3}{z^6}\) find \(\displaystyle \frac{{{\partial ^6}f}}{{\partial y\partial {z^2}\partial y\partial {x^2}}}\). Solution
6. Given \(w = {u^2}{{\bf{e}}^{ - 6v}} + \cos \left( {{u^6} - 4u + 1} \right)\) find \({w_{v\,u\,u\,v\,v}}\). Solution
7. Given \(G\left( {x,y} \right) = {y^4}\sin \left( {2x} \right) + {x^2}{\left( {{y^{10}} - \cos \left( {{y^2}} \right)} \right)^7}\) find \({G_{y\,y\,y\,x\,x\,x\,y}}\). Solution