Let

*be any function and*

**y=f(x)***exists at*

**dy/dx**

**x=a**Then at the corresponding point

*of the function tangent to the curve exists and if the tangent*

**[a , f(a)]**makes an angle

*with the positive direction of*

**theta***we have*

**x-axis***at*

**tan(theta)=dy/dx**

**x=a .**

**tan(theta) = gradient of the tangent to the curve***at*

**y=f(x)**

**x=a**therefore ; if

*is differentiable at*

**f(x)***then tangent to the curve at*

**x=a***must exist and it must be unique .*

**x=a**In any graph of

*tangent is not unique at*

**/x/***x=0*.

i.e. ; tangent at point

*when*

**x=0***from left is not same as the tangent*

**x-->0***when*

**x=0***from right .*

**x-->0**when

*from left , gradient of the tangent is*

**x-->0***and when*

**tan135degree=-1***from right , gradient of the tangent is*

**x-->0**

**tan45degree=1.**Hence tangent is not unique at

*and consequently*

**x=0***is not differentiable at*

**/x/***.*

**x=0**
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