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Section 13.2 : Partial Derivatives
For problems 1 – 8 find all the 1st order partial derivatives.
- \(f\left( {x,y,z} \right) = 4{x^3}{y^2} - {{\bf{e}}^z}{y^4} + \frac{{{z^3}}}{{{x^2}}} + 4y - {x^{16}}\) Solution
- \(w = \cos \left( {{x^2} + 2y} \right) - {{\bf{e}}^{4x - {z^{\,4}}y}} + {y^3}\) Solution
- \(f\left( {u,v,p,t} \right) = 8{u^2}{t^3}p - \sqrt v \,{p^2}{t^{ - 5}} + 2{u^2}t + 3{p^4} - v\) Solution
- \(f\left( {u,v} \right) = {u^2}\sin \left( {u + {v^3}} \right) - \sec \left( {4u} \right){\tan ^{ - 1}}\left( {2v} \right)\) Solution
- \(\displaystyle f\left( {x,z} \right) = {{\bf{e}}^{ - x}}\sqrt {{z^4} + {x^2}} - \frac{{2x + 3z}}{{4z - 7x}}\) Solution
- \(g\left( {s,t,v} \right) = {t^2}\ln \left( {s + 2t} \right) - \ln \left( {3v} \right)\left( {{s^3} + {t^2} - 4v} \right)\) Solution
- \(\displaystyle R\left( {x,y} \right) = \frac{{{x^2}}}{{{y^2} + 1}} - \frac{{{y^2}}}{{{x^2} + y}}\) Solution
- \(\displaystyle z = \frac{{{p^2}\left( {r + 1} \right)}}{{{t^3}}} + pr\,{{\bf{e}}^{2p + 3r + 4t}}\) Solution
- Find \(\displaystyle \frac{{\partial z}}{{\partial x}}\) and \(\displaystyle \frac{{\partial z}}{{\partial y}}\) for the following function. \[{x^2}\sin \left( {{y^3}} \right) + x{{\bf{e}}^{3z}} - \cos \left( {{z^2}} \right) = 3y - 6z + 8\] Solution