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.
- \({z^4} - {y^2} + {x^2} = 6{x^2}{y^3}{z^7}\)
- \({x^2}\sin \left( z \right) + \left( {{x^2} - 1} \right){y^4} = {z^6}\)