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 $x$?