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Section 13.4 : Higher Order Partial Derivatives
5. Find all 2nd order derivatives for the following function.
\[h\left( {x,y} \right) = {{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} - \frac{{{y^3}}}{x}\]Show All Steps Hide All Steps
Start SolutionFirst, we know we’ll need the two 1st order partial derivatives. Here they are,
\[{h_x} = 4{x^3}{y^6}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} + \frac{{{y^3}}}{{{x^2}}}\hspace{0.5in}{h_y} = 6{y^5}{x^4}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} - \frac{{3{y^2}}}{x}\] Show Step 2Now let’s compute each of the second order partial derivatives.
\[\begin{align*}{h_{x\,x}} & = {\left( {{h_x}} \right)_x} = 12{x^2}{y^6}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} + 16{x^6}{y^{12}}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} - \frac{{2{y^3}}}{{{x^3}}}\\ {h_{x\,y}} & = {\left( {{h_x}} \right)_y} = 24{x^3}{y^5}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} + 24{x^7}{y^{11}}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} + \frac{{3{y^2}}}{{{x^2}}}\\ {h_{y\,x}} & = {h_{x\,y}} = 24{x^3}{y^5}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} + 24{x^7}{y^{11}}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} + \frac{{3{y^2}}}{{{x^2}}}\hspace{0.5in}{\mbox{by Clairaut's Theorem}}\\ {h_{y\,y}} & = {\left( {{h_y}} \right)_y} = 30{y^4}{x^4}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} + 36{y^{10}}{x^8}{{\bf{e}}^{{x^{\,4}}{y^{\,6}}}} - \frac{{6y}}{x}\end{align*}\]