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Section 13.4 : Higher Order Partial Derivatives

For problems 1 – 3 verify Clairaut’s Theorem for the given function.

1. \(Q\left( {s,t} \right) = \ln \left( {st} \right) - {s^4}\sin \left( {6t} \right) + st\)
2. \(f\left( {u,w} \right) = \sin \left( {uw} \right) + 4{u^2}{w^{ - 2}}\)
3. \(f\left( {x,y} \right) = {{\bf{e}}^{x\,y}}\sin \left( y \right)\)

For problems 4 – 9 find all 2nd order derivatives for the given function.

1. \(h\left( {x,y} \right) = {x^4}{y^{ - 2}} - 4xy + {{\bf{e}}^{7y}} + \ln \left( {2x} \right)\)
2. \(\displaystyle A\left( {u,v} \right) = {u^2}\cos \left( {3v} \right) + \ln \left( {\frac{u}{{4{v^2}}}} \right)\)
3. \(g\left( {v,w} \right) = \ln \left( {v\sin \left( w \right)} \right) + \sin \left( {v\ln \left( w \right)} \right)\)
4. \(f\left( {x,y} \right) = \cos \left( {{x^2} + {y^2}} \right) - \sin \left( {xy} \right)\)
5. \(h\left( {x,y,z} \right) = 7{x^3}{y^2}{z^4} + 8\)
6. \(\displaystyle Q\left( {u,v,w} \right) = {u^4}\sin \left( {{w^2}} \right) - \frac{{2v}}{{{u^4}}} + \ln \left( {{v^2}w} \right)\)

For problems 10 & 11 find all 3rd order derivatives for the given function.

1. \(h\left( {x,y} \right) = {x^4}{y^5} - 5\sqrt x + 8{y^2}\)
2. \(\displaystyle A\left( {u,v} \right) = {u^3}\sin \left( {2v} \right) - \frac{{{u^3}}}{{{v^2}}}\)
3. Given \(f\left( {x,y,z} \right) = {{\bf{e}}^{ - z}}\cos \left( {4y} \right)\ln \left( {2x} \right)\) find \({f_{x\,y\,y\,z\,x\,z}}\).
4. Given \(w = \ln \left( {\frac{{xy}}{z}} \right) + 8{x^4}{y^3}\sqrt z \) find \(\displaystyle \frac{{{\partial ^5}w}}{{\partial x\partial {z^2}\partial y\partial x}}\).
5. Given \(\displaystyle h\left( {u,v} \right) = \cos \left( {{u^4} + {u^2} + 1} \right) - \frac{{\sqrt u }}{{{v^3}}}\) find \(\displaystyle \frac{{{\partial ^7}h}}{{\partial {u^2}\partial v\partial {u^4}}}\).
6. Given \(\displaystyle f\left( {x,y} \right) = \frac{{{x^6}}}{{1 + 6y}} - \cos \left( {{x^2}} \right) + 6{{\bf{e}}^x}\sin \left( y \right)\) find \({f_{x\,x\,y\,x\,y\,x}}\).