# Chapter 13: Derivatives as a Function and One-sided Derivatives

## Derivative as Function

Recall from last lecture,

The SLOPE of ﻿$f(x)$﻿ at ﻿$x=x_0$﻿ is

﻿$f'(x_0) = \lim_ {h \rightarrow 0} \frac{f(x_0 +h)-f(x_0)} {h}$﻿

﻿$= \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{ x-x_0}$﻿

This function of ﻿$x_0$﻿ is ''derived ' from ﻿$f(x)$﻿ and we call it the DERIVATIVE of ﻿$f(x)$﻿

### Example

Stretch the derivative of ﻿$f(x)$﻿

#### Example

Calculate ﻿$f'(x)$﻿ if ﻿$f(x) = x^2$﻿

﻿$f'(x)= \lim_{h \rightarrow 0} \frac{f(x+h) -f(x) }{h} = \lim_{h \rightarrow 0} \frac{ (x+h)2-(x)^2}{h}$﻿

﻿$=\lim_{h \rightarrow 0} \frac {\cancel{x^2}+2xh+h^2 \cancel{-x^2}}{h}$﻿

﻿$= \lim_{h \rightarrow 0}$﻿ ﻿$2x+h$﻿

﻿$=2x$﻿

#### Notation

If ﻿$y=f(x)$﻿ we write:

'"Leibnitz" ﻿$\frac{dy}{dx} = \frac {df(x)}{d(x)} = f'(x) = y'$﻿ "Newton"

#### Example

Find ﻿$f'(x)$﻿ if ﻿$f(x) = \frac{1}{x}$﻿

﻿$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) -f(x)}{h}$﻿

﻿$= \lim_{h \rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}$﻿ ﻿$\rightarrow$﻿ Clear denominators

﻿$= \lim_{h \rightarrow 0} \frac{x-(x+h)}{h \cdot x \cdot (x+h)}$﻿

﻿$= \lim_{h \rightarrow 0} \frac{-h}{h \cdot x \cdot (x+h)}$﻿

﻿$= \lim_{h \rightarrow 0} \frac{-1}{x(x+h)}$﻿

﻿$=-\frac{1}{x^2}$﻿

#### Example

Find ﻿$f'(x)$﻿ if ﻿$f(x) =\sqrt{x}$﻿

﻿$f'(x) = \lim_{h \rightarrow 0} \frac{\sqrt{x+h }-\sqrt{x}}{h} \rightarrow$﻿ Conjugate the top

﻿$=$﻿ ﻿$\lim_{h \rightarrow 0} \frac{\sqrt{x+h }-\sqrt{x}}{h}[\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}]$﻿

﻿$=\lim_{h \rightarrow 0} \frac{x+h-x}{h[\sqrt{x+h}+\sqrt{x}]}$﻿

﻿$=\lim_{h \rightarrow 0} \frac{1}{\sqrt{x+h}+\sqrt{x}}$﻿

﻿$=\frac{1}{2\sqrt{x}}$﻿

## One Sided

#### DEfinition

The RIGHT HAND DERIVATIVE of ﻿$f(x)$﻿ at ﻿$x=x_0$﻿

﻿$f'_+(x_0)= \lim_{h \rightarrow 0^+} \frac{f(x_0+h)-f(x_0)} {h}$﻿

﻿$= \lim_{ x \rightarrow x_0^+ } \frac{f(x) -f(x_0)}{x-x_0}$﻿

The LEFT HAND DERIVATIVE is similar

#### Example

If ﻿$f(x) =|x|$﻿ find ﻿$f'_+(0)$﻿ and ﻿$f'_-(0)$﻿

﻿$f'_+(0)= \lim _{h \rightarrow 0^+} \frac{|0+h|-|0|}{h} = \lim_{h \rightarrow 0^+} \frac{ |h|}{h} = +1$﻿

﻿$f'_-(0)= \lim_{ h \rightarrow 0^+} \frac{|h|}{h} =-1$﻿

#### Definition

If ﻿$f'_+ (x) =f'_- (x)$﻿ then ﻿$f$﻿ is DIFFERENTIABLE at ﻿$x$﻿

#### Fact

If ﻿$f$﻿ is differentiable at ﻿$x$﻿ then it is continuous at ﻿$x$﻿

﻿$*$﻿ We have differentiable ﻿$\Rightarrow$﻿ continuous but not continuous ﻿$\Rightarrow$﻿ differentiable

#### Example

﻿$|x|$﻿ is continuous but not differentiable at ﻿$x =0$﻿

Q: If ﻿$f'_+ (x) \neq f'_- (x)$﻿ then what is the ''slope'' at ﻿$x$﻿?