2.2.74 Derivatives and differentials
Condition: Let \( \quad u=\frac{1}{x}(\varphi(x-y)+\psi(x+y)) \), where \( \varphi \) and \( \psi \) are differentiable functions. Show that \[ \frac{\partial}{\partial x}\left(x^{2} \frac{\partial u}{\partial x}\right)=x^{2} \frac{\partial^{2} u}{\partial y^{2}} \]
Calculation of derivatives and differentials of first and higher orders of functions of one and many variables, including partial derivatives.