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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 13.2 : Partial Derivatives
For problems 1 – 13 find all the 1st order partial derivatives.
- \(f\left( {x,y,z} \right) = {x^3}\sqrt y + 4{z^3}{y^2} - xyz + {x^2} - \sqrt[3]{{{z^5}}}\)
- \(W\left( {a,b,c,d} \right) = {a^2} + {b^3} - {c^2}{d^4} - 5{a^3}c - {d^6}{b^2}a\)
- \(\displaystyle A\left( {p,t,u} \right) = \frac{1}{{p\,{t^2}}} - \frac{{{t^3}}}{{{u^2}}} + \frac{{4u\,p}}{{{t^4}}}\)
- \(g\left( {x,y,z} \right) = \sqrt {{x^2} + {z^{ - 2}}} + \sin \left( {xy - {x^2}} \right)\)
- \(f\left( {s,t} \right) = \cos \left( {s{{\bf{e}}^{{t^{\,2}}}}} \right) + \cos \left( {s + {{\bf{e}}^{{t^{\,2}}}}} \right)\)
- \(\displaystyle f\left( {x,y} \right) = \ln \left( {\frac{y}{x}} \right) + \ln \left( {\frac{1}{{x + y}}} \right) - \ln \left( {\frac{x}{6}} \right)\)
- \(\displaystyle A\left( {y,z} \right) = \frac{1}{{y - 4{z^5}}} + \tan \left( {y{z^2} - {y^3}} \right)\)
- \(\displaystyle g\left( {u,v} \right) = \frac{u}{v}\cos \left( {\frac{v}{u}} \right) + 4u - {v^2}u\)
- \(w = \left( {x - y} \right){{\bf{e}}^{4x + {z^{\,6}}}} - \sin \left( {2x + 7z} \right)\sec \left( {y{z^3}} \right)\)
- \(f\left( {u,v,w} \right) = \left( {uw + 4} \right){\sin ^{ - 1}}\left( {{u^2} + {v^2}} \right) - \ln \left( {\frac{{{w^2}}}{{{v^4}}}} \right)\)
- \(\displaystyle f\left( {x,y,z} \right) = \sin \left( {\frac{z}{{{z^2} + x}}} \right) - \frac{{6{x^2} + y}}{{{y^2} - {z^2}}}\)
- \(\displaystyle g\left( {s,t,p} \right) = \frac{{{p^3}{t^2}}}{{{s^2} + 1}} + \frac{{\left( {4s - 1} \right){t^2}}}{{6 - s}}\)
- \(f\left( {x,y,z,w} \right) = {x^2}\sin \left( {4y} \right) + {z^3}\left( {6x - y} \right) + {y^4}\)
For problems 14 & 15 find \(\displaystyle \frac{{\partial z}}{{\partial x}}\) and \(\displaystyle \frac{{\partial z}}{{\partial y}}\) for the given function.
- \({z^4} - {y^2} + {x^2} = 6{x^2}{y^3}{z^7}\)
- \({x^2}\sin \left( z \right) + \left( {{x^2} - 1} \right){y^4} = {z^6}\)