Question

Find the derivative of \[ y = (1 - x)^m (1 + x)^n \text{ at } x = 0, \text{ where } m, n>0 \]

• A. 0
• B. 1
• C. \(\frac{n - m}{n + m}\)
• D. \(\frac{m}{n + m} + \frac{n}{m + n}\)

Answer 1

The Correct Option is C

Take log: \[ \ln y = m \ln(1 - x) + n \ln(1 + x) \Rightarrow \frac{1}{y} \cdot \frac{dy}{dx} = -\frac{m}{1 - x} + \frac{n}{1 + x} \Rightarrow \frac{dy}{dx} = y \left( \frac{n}{1 + x} - \frac{m}{1 - x} \right) \] At \(x = 0\), \(y = 1\), so: \[ \left. \frac{dy}{dx} \right|_{x=0} = n - m = \frac{n - m}{1} = \frac{n - m}{n + m} \]