Differentiability
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It fails to be differentiable if:  It fails to be differentiable if:  
  *<math>f(x)</math> is not continuous at <math>a</math>  +  *<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]]  {{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]]  
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  *The graph has a sharp corner at <math>a</math>  +  *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]]  {{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]]  
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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.}}  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  +  *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]]  {{hideConsider the function <math>f(x)=x^{1 \over\ 3}</math>. Plotting the graph: [[Image: math images 7.jpgleftthumb]]  
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FieldLinks=Visit this site for an interactive experience with differentiability:  FieldLinks=Visit this site for an interactive experience with differentiability:  
:*http://wwwmath.mit.edu/18.013A/HTML/chapter06/section01.html  :*http://wwwmath.mit.edu/18.013A/HTML/chapter06/section01.html  
+  }  
+  Field=Algebra  
+  InProgress=No  
}}  }} 
Current revision
 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 
