Question-24

Consider two functions \(f:\mathbb{R}\rightarrow \mathbb{R}\) and \(g:\mathbb{R}\rightarrow ( 1,\infty )\). Both functions are differentiable at a point \(c\). Which of the following functions is/are ALWAYS differentiable at \(c\)? The symbol $$ denotes product and the symbol $$ denotes composition of functions.

• \(f\pm g\)

• \(f\cdot g\)

• \(\cfrac{f}{g}\)

• \(f\circ g+g\circ f\)

NoteAnswer

• \(f\pm g\)

• \(f\cdot g\)

• \(\cfrac{f}{g}\)

• \(f\circ g+g\circ f\)

NoteSolution

• The sum of two functions that are differentiable at a point are also differentiable.

• The product rule shows that \(f\cdot g\) is differentiable at \(c\).

• The quotient rule shows that \(\cfrac{f}{g}\) is differentiable at \(c\). Also note that the quotient rule is applicable since \(g( c) \neq 0\).

• \(f\circ g\) or \(g\circ f\) may not be differentiable at \(c\). For example, consider \(g( x) =1+\cfrac{1}{1+|x|}\) and \(f( x) =x-2\). Then, we see that \(( g\circ f)( x) =1+\cfrac{1}{1+|x-2|}\). Now, for \(c=2\), both \(g\) and \(f\) are differentiable. However, \(g\circ f\) isn’t.