Consider two functions \(f : \mathbb{R} \to \mathbb{R}\) and \(g : \mathbb{R} \to (1, \infty)\). Both functions are differentiable at a point \(c\). Which of the following functions is/are ALWAYS differentiable at \(c\)? The symbol \(\cdot\) denotes product and the symbol \(\circ\) denotes composition of functions.
Answer
(A), (B), (C)
Explanation
Options A, B and C use only the values and derivatives of \(f\) and \(g\) at \(c\):
- sums, differences and products of functions differentiable at \(c\) are differentiable at \(c\);
- the quotient rule for \(\frac{f}{g}\) also needs \(g(c) \neq 0\), which holds because \(g\) takes values in \((1, \infty)\).
Option D is different: \(f \circ g\) needs \(f\) to be differentiable at \(g(c)\), and \(g \circ f\) needs \(g\) to be differentiable at \(f(c)\). Neither is given. For example, take \(c = 0\), \(f(x) = |x - 2|\) and \(g(x) = 1 + e^{x}\) (both differentiable at 0, and \(g > 1\)). Then \(f(g(x)) = |e^{x} - 1|\) has a corner at 0 while \(g(f(x))\) is smooth there, so the sum is not differentiable at 0.
Answer: A, B, C