Section 13.4 : Higher Order Partial Derivatives
2. Verify Clairaut’s Theorem for the following function.
\[A\left( {x,y} \right) = \cos \left( {\frac{x}{y}} \right) - {x^7}{y^4} + {y^{10}}\]Show All Steps Hide All Steps
Start SolutionFirst, we know we’ll need the two 1st order partial derivatives. Here they are,
\[{A_x} = - \frac{1}{y}\sin \left( {\frac{x}{y}} \right) - 7{x^6}{y^4}\hspace{0.5in}{A_y} = \frac{x}{{{y^2}}}\sin \left( {\frac{x}{y}} \right) - 4{x^7}{y^3} + 10{y^9}\] Show Step 2Now let’s compute each of the mixed second order partial derivatives.
\[\begin{align*}{A_{x\,y}} & = {\left( {{A_x}} \right)_y} = \frac{1}{{{y^2}}}\sin \left( {\frac{x}{y}} \right) + \frac{x}{{{y^3}}}\cos \left( {\frac{x}{y}} \right) - 28{x^6}{y^3}\\ {A_{y\,x}} & = {\left( {{A_y}} \right)_x} = \frac{1}{{{y^2}}}\sin \left( {\frac{x}{y}} \right) + \frac{x}{{{y^3}}}\cos \left( {\frac{x}{y}} \right) - 28{x^6}{y^3}\end{align*}\]Okay, we can see that \({A_{x\,y}} = {A_{y\,x}}\) and so Clairaut’s theorem has been verified for this function.