Differentiation：練習問題

Find \(\frac{dy}{dx}\) for the function \(y = \tanh^{-1}(\sin x)\), where \(|\sin x| < 1\).

Derive the Maclaurin series for \(f(x) = \tan^{-1}(e^x)\) up to and including the term in \(x^3\).

A curve is defined implicitly by the equation \(x^2 + \sinh y = 1\). Find the value of \(\frac{d^2y}{dx^2}\) at the point where \(x = 0\).

A curve is defined by the parametric equations \(x = \ln(t + 1)\) and \(y = \sinh(2t)\). Determine the value of \(\frac{d^2y}{dx^2}\) when \(t = 0\).

Given the function \(f(x) = \tanh^{-1}(x)\), use successive differentiation to find the Maclaurin series for \(f(x)\) up to and including the term in \(x^3\). Justify your answer by showing the values of \(f(0)\), \(f'(0)\), \(f''(0)\), and \(f'''(0)\).

(a) A curve is defined parametrically by the equations \(x = \ln(\cosh t)\) and \(y = t^2\).

Find \(\frac{dy}{dx}\) in terms of \(t\).

(b) Hence, find \(\frac{d^2y}{dx^2}\) in terms of \(t\), simplifying your answer.