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

Derivative as Function

Recall from last lecture,

The SLOPE of f(x)f(x) at x=x0x=x_0 is


f(x0)=limh0f(x0+h)f(x0)hf'(x_0) = \lim_ {h \rightarrow 0} \frac{f(x_0 +h)-f(x_0)} {h}


=limxx0f(x)f(x0)xx0= \lim_{x \rightarrow x_0} \frac{f(x) - f(x_0)}{ x-x_0}

This function of x0x_0 is ''derived ' from f(x)f(x) and we call it the DERIVATIVE of f(x)f(x)


Example

Stretch the derivative of f(x)f(x)


Example

Calculate f(x)f'(x)  if f(x)=x2f(x) = x^2

f(x)=limh0f(x+h)f(x)h=limh0(x+h)2(x)2hf'(x)= \lim_{h \rightarrow 0} \frac{f(x+h) -f(x) }{h} = \lim_{h \rightarrow 0} \frac{ (x+h)2-(x)^2}{h}

=limh0x2+2xh+h2x2h=\lim_{h \rightarrow 0} \frac {\cancel{x^2}+2xh+h^2 \cancel{-x^2}}{h}

=limh0= \lim_{h \rightarrow 0} 2x+h2x+h

=2x=2x


Notation

If y=f(x)y=f(x) we write:

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


Example

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

f(x)=limh0f(x+h)f(x)hf'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) -f(x)}{h}

=limh01x+h1xh= \lim_{h \rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h} \rightarrow  Clear denominators

=limh0x(x+h)hx(x+h)= \lim_{h \rightarrow 0} \frac{x-(x+h)}{h \cdot x \cdot (x+h)}

=limh0hhx(x+h)= \lim_{h \rightarrow 0} \frac{-h}{h \cdot x \cdot (x+h)}

=limh01x(x+h)= \lim_{h \rightarrow 0} \frac{-1}{x(x+h)}

=1x2=-\frac{1}{x^2}


Example

Find f(x)f'(x) if f(x)=xf(x) =\sqrt{x}

f(x)=limh0x+hxhf'(x) = \lim_{h \rightarrow 0} \frac{\sqrt{x+h }-\sqrt{x}}{h} \rightarrow Conjugate the top

== limh0x+hxh[x+h+xx+h+x]\lim_{h \rightarrow 0} \frac{\sqrt{x+h }-\sqrt{x}}{h}[\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}]

=limh0x+hxh[x+h+x]=\lim_{h \rightarrow 0} \frac{x+h-x}{h[\sqrt{x+h}+\sqrt{x}]}

=limh01x+h+x=\lim_{h \rightarrow 0} \frac{1}{\sqrt{x+h}+\sqrt{x}}

=12x=\frac{1}{2\sqrt{x}}

One Sided

DEfinition

The RIGHT HAND DERIVATIVE of f(x)f(x)  at x=x0 x=x_0

f+(x0)=limh0+f(x0+h)f(x0)hf'_+(x_0)= \lim_{h \rightarrow 0^+} \frac{f(x_0+h)-f(x_0)} {h}

=limxx0+f(x)f(x0)xx0= \lim_{ x \rightarrow x_0^+ } \frac{f(x) -f(x_0)}{x-x_0}

The LEFT HAND DERIVATIVE is similar


Example

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


f+(0)=limh0+0+h0h=limh0+hh=+1f'_+(0)= \lim _{h \rightarrow 0^+} \frac{|0+h|-|0|}{h} = \lim_{h \rightarrow 0^+} \frac{ |h|}{h} = +1

f(0)=limh0+hh=1f'_-(0)= \lim_{ h \rightarrow 0^+} \frac{|h|}{h} =-1


Definition

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


Fact

If ff is differentiable at xx then it is continuous at xx

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


Example

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


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


































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