Differentiability
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+  {{Image Description  
+  ImageName=A Differentiable Function  
+  Image=Math images 4.jpg  
+  ImageIntro=<math> y=x^2</math> is an example of a function that is differentiable everywhere.  
+  ImageDescElem=If we were to draw a tangent through every point on the curve in the main image, we would not encounter any difficulty at any point because there are no discontinuities, sharp corners and straight vertical portions at any point. This means that the function is '''differentiable'''.  
+  ImageDesc=A function is differentiable at a point if it has a tangent at every point. That is, a function is differentiable at <math>x</math> if the [[Limitlimit]]  
+  
+  <math> \lim_{a \to 0} {{f(x+a)f(x)} \over\ a} </math> exists.  
+  
+  It fails to be differentiable if:  
+  
+  *<math>f(x)</math> is not continuous at <math>a</math>  
+  
+  {{hide[[Image:math_images_5.jpgleftthumb The limit does not approach the same value from the right as it does from the left, so the function is not differentiable]]  
+  
+  
+  For example <math> f(x)= {1 \over\ x} </math> is not continuous at <math>x=0</math>. The function is undefined at that point, hence it is not differentiable.  
+  Computing the limit:  
+  
+  
+  <math>\lim_{a \to 0} {{f(a)f(0)} \over\ a}= \lim_{a \to 0} {{{1 \over\ a} 0} \over\ a}</math>  
+  <math>= \lim_{a \to 0} {1 \over\ a^2}</math>  
+  
+  
+  As <math>a</math> approaches <math>0</math> from the left and right,the denominator becomes smaller and smaller, hence the limit approaches <math> \infty</math> and <math> \infty</math> respectively. The limits from the right and left are different so the limit does not exist hence the function is not differentiable.}}  
+  
+  
+  *The graph has a sharp corner at <math>a</math>  
+  
+  {{hideThe function <math>f(x)=x</math> has a sharp corner at <math>x=0</math>. [[Image:Absolute_value.pngrightthumb http://commons.wikimedia.org/wiki/File:Absolute_value.png]]  
+  
+  Computing the limit:  
+  <math>\lim_{a \to 0} {{f(a)f(0)} \over\ a}= \lim_{a \to 0} {{a0} \over\ a}  
+  = \lim_{a \to 0} {{a} \over\ a}</math>  
+  
+  As <math>a</math> approaches <math>0</math> from the right, the ratio is <math>1</math>, from the left, the ratio is <math>1</math>. The limits are different, so the function is not differentiable.}}  
+  
+  *The graph has a vertical tangent line  
+  
+  {{hideConsider the function <math>f(x)=x^{1 \over\ 3}</math>. Plotting the graph: [[Image: math images 7.jpgleftthumb]]  
+  
+  If you take the <math> \lim_{a \to 0} {{f(a)f(0)} \over\ a}= \lim_{a \to 0} {a^{1 \over\ 3} \over\ a} = \lim_{a \to 0} {1 \over\ a^{2 \over\ 3}}</math>  
+  
+  As <math>a</math> approaches infinity, the denominator becomes smaller and smaller, so the function grows beyond bounds, hence no derivative at <math>x=0</math>. The function is therefore not differentiable.}}  
+  
+  
+  
+  
+  '''Note: While all differentiable functions are continuous, all continuous functions may not be differentiable.'''  
+  PreK=No  
+  Elementary=No  
+  MiddleSchool=No  
+  HighSchool=No  
+  AuthorName=Lizah Masis  
+  SiteName=made with Mathematica  
+  SiteURL=http://wikis.swarthmore.edu/miwiki/index.php/User:Lmasis1  
+  Field=Calculus  
+  FieldLinks=Visit this site for an interactive experience with differentiability:  
+  :*http://wwwmath.mit.edu/18.013A/HTML/chapter06/section01.html  
+  }  
+  Field=Algebra  
+  InProgress=No  
+  }}  
{{HelperPage1=Roulette2=Strange Attractors3=Critical Points4=Taylor Series}}  {{HelperPage1=Roulette2=Strange Attractors3=Critical Points4=Taylor Series}}  
{{Image Description  {{Image Description 
Revision as of 12:49, 2 July 2013
 is an example of a function that is differentiable everywhere.
A Differentiable Function 

Contents 
Basic Description
If we were to draw a tangent through every point on the curve in the main image, we would not encounter any difficulty at any point because there are no discontinuities, sharp corners and straight vertical portions at any point. This means that the function is differentiable.A More Mathematical Explanation
A function is differentiable at a point if it has a tangent at every point. That is, a function is di [...]
A function is differentiable at a point if it has a tangent at every point. That is, a function is differentiable at if the limit
exists.
It fails to be differentiable if:
 is not continuous at
 The graph has a sharp corner at
 The graph has a vertical tangent line
Note: While all differentiable functions are continuous, all continuous functions may not be differentiable.
Teaching Materials
 There are currently no teaching materials for this page. Add teaching materials.
Related Links
Additional Resources
Visit this site for an interactive experience with differentiability:
}
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
This is a Helper Page for:


Roulette 
Strange Attractors 
Critical Points 
Taylor Series 
 is an example of a function that is differentiable everywhere.
Basic Description
If we were to draw a tangent through every point on the curve in the main image, we would not ecounter any difficulty at any point because there are no discontinuities, sharp corners and straight vertical portions at any point. This means that the function is differentiable.A More Mathematical Explanation
A function is differentiable at a point if it has a tangent at every point. That is, a function is di [...]
A function is differentiable at a point if it has a tangent at every point. That is, a function is differentiable at if the limit
exists.
It fails to be differentiable if:
 is not continuous at
 The graph has a sharp corner at
 The graph has a vertical tangent line
Note: While all differentiable functions are continuous, all continuous functions may not be differentiable.
Teaching Materials
 There are currently no teaching materials for this page. Add teaching materials.
Related Links
Additional Resources
Visit this site for an interactive experience with differentiability:
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
A Differentiable Function 
