📝 题目
例 7 求函数 $u = f\left( {{xy},\frac{y}{x}}\right)$ 的二阶偏导数.
💡 答案与解析
解 先求一阶偏导数:
$$ {u}_{x}^{\prime } = y{f}_{1}^{\prime } - \frac{y}{{x}^{2}}{f}_{2}^{\prime },\;{u}_{y}^{\prime } = x{f}_{1}^{\prime } + \frac{1}{x}{f}_{2}^{\prime }. $$
再求二阶偏导数:
$$ {u}_{xx}^{\prime \prime } = {y}^{2}{f}_{11}^{\prime \prime } - 2\frac{{y}^{2}}{{x}^{2}}{f}_{12}^{\prime \prime } + \frac{{y}^{2}}{{x}^{4}}{f}_{22}^{\prime \prime } + \frac{2y}{{x}^{3}}{f}_{2}^{\prime }, $$
$$ {u}_{xy}^{\prime \prime } = {xy}{f}_{11}^{\prime \prime } - \frac{y}{{x}^{3}}{f}_{22}^{\prime \prime } + {f}_{1}^{\prime } - \frac{1}{{x}^{2}}{f}_{2}^{\prime }, $$
$$ {u}_{yy}^{\prime \prime } = {x}^{2}{f}_{11}^{\prime \prime } + 2{f}_{12}^{\prime \prime } + \frac{1}{{x}^{2}}{f}_{22}^{\prime \prime }. $$
评注 对一阶偏导数再求一次偏导数时, 必须注意每项都要微到, 每项中每个因子都要微到, 每个因子中每个变量都要微到. 如求 ${u}_{xx}^{\prime \prime }$ 时丢了 $\frac{2y}{{x}^{3}}{f}_{2}^{\prime }$ 这一项,说明对因子 $\frac{y}{{x}^{2}}$ 没有微到. 如丢了 ${f}_{12}^{\prime \prime }$ 项,说明
${f}_{1}^{\prime }$ 中对第二个变量没有微到.