Problem Set - Differentiability

## Problem Set - Differentiability

Last updates: 7 December 2009

## Differentiability

 Let $a,b\in ℝ$ and let $f:\left[a,b\right]\to ℝ$ be a function. Let $c\in \left[a,b\right]$ and carefully define $f\prime \left(c\right)$. Prove that if $f:\left[a,b\right]\to ℝ$ and $g:\left[a,b\right]\to ℝ$ are functions then $\left(fg\right)\prime \left(c\right)=f\left(c\right)g\prime \left(c\right)+f\prime \left(c\right)g\left(c\right)$, whenever $f\prime \left(c\right)$ and $g\prime \left(c\right)$ exist. Let $f:{ℝ}_{>0}\to ℝ$ be such that $f$ is differentiable at $x=1$ and if $x,y\in {ℝ}_{>0}$ then $f\left(xy\right)=f\left(x\right)+f\left(y\right)$. Show that (a)   if $c\in {ℝ}_{>0}$ then $f$ is differentiable at $x=c$, (b)   if $c\in {ℝ}_{>0}$ then $f\prime \left(c\right)=f\prime \left(1\right)/c$, (c)   Show that $f$ is infinitely differentiable. Let $f:ℝ\to ℝ$ be such that $f$ is differentiable at $x=0$ and if $x,y\in ℝ$ then $f\left(x+y\right)=f\left(x\right)f\left(y\right)$. Show that (a)   if $c\in ℝ$ then $f$ is differentiable at $x=c$, (b)   if $c\in {ℝ}_{>0}$ then $f\prime \left(c\right)=f\prime \left(0\right)f\left(c\right)$, (c)   Show that $f$ is infinitely differentiable. Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\left\{\begin{array}{cc}-{x}^{2},& \text{if}\phantom{\rule{0.5em}{0ex}}x\le 0,\hfill \\ x,& \text{if}\phantom{\rule{0.5em}{0ex}}x>0.\hfill \end{array}$ Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$? Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\left\{\begin{array}{cc}-{x}^{2},& \text{if}\phantom{\rule{0.5em}{0ex}}x\le 0,\hfill \\ {x}^{3},& \text{if}\phantom{\rule{0.5em}{0ex}}x>0.\hfill \end{array}$ Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$? Let $f:ℝ\to ℝ$ be given by $f\left(x\right)=\left\{\begin{array}{cc}\frac{\mathrm{sin}x}{x},& \text{if}\phantom{\rule{0.5em}{0ex}}x<0,\hfill \\ 1+{x}^{2},& \text{if}\phantom{\rule{0.5em}{0ex}}x\ge 0.\hfill \end{array}$ Is $f$ continuous at $x=0$? Is $f$ differentiable at $x=0$? Let $a,b\in ℝ$ and assume that $f:\left[a,b\right)\to ℝ$ is differentiable on $\left(a,b\right)$ and continuous on $\left[a,b\right)$. Assume that the limit $\underset{x\to a+}{\mathrm{lim}}f\prime \left(x\right)=L$ exists. Prove that the right derivative ${f}_{+}\prime \left(a\right)$ exists and that ${f}_{+}\prime \left(a\right)=L$. Let $a,b\in ℝ$ and assume that $f:\left(a,b\right)\to ℝ$ is differentiable at $c$. Show that $\underset{h\to 0+}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c-h\right)}{2h}$ exists and equals $f\prime \left(c\right)$. Is the converse true? Prove that $\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}$. Prove that $\frac{d}{dx}\mathrm{arcsin}x=\frac{1}{\sqrt{1-{x}^{2}}}$.