If \(y=a+x\log\frac{y}{b}\), find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) when \(x\) is zero. \par Shew that, if \(x\) is so small that the value of \(x^3\) and of higher powers may be neglected, then \[ y = a+x\log\frac{a}{b} + \frac{x^2}{a}\log\frac{a}{b}. \]
Prove Leibnitz's rule for the repeated differentiation of the product of two functions of \(x\). \par Prove that \[ \left(\frac{d}{dx}\right)^n \frac{\log x}{x} = (-1)^n\frac{n!}{x^{n+1}}\left(\log x - 1 - \frac{1}{2}-\dots-\frac{1}{n}\right). \]
Prove that for a curve, the radius of curvature \(\frac{ds}{d\psi}\) is equal to \[ \left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{\frac{3}{2}} / \frac{d^2y}{dx^2}. \] Prove that the length of the radius of curvature of the curve \(y=\frac{x^3}{3}-x\) is a minimum at points for which \(x=\pm 1.07\), approximately.