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Existence of derivative

Right hand and left hand derivatives
  • The right-hand derivative of f at x = a denoted by f′(a+) is defined by
     
    f′(a+) = 87847.png
     
    provided the limit exists and is finite. When h 0, the point B moving along the curve tends to A, i.e., B A then the chord AB approaches the tangent line AT at the point A and then φ ψ
     
     f′(a+) = 87841.png = tan ψ
  • The left-hand derivative of f at x = a denoted by f ′(a) is defined by
     
    f′(a) = 87829.png
     
    Provided the limit exists and is finite. When h 0, the point C moving along the curve tends to A, i.e., C A then the chord CA approaches the tangent line AT at the point A then
     
    f′(a) = 87823.png
  • At A, f(x) is differentiable if f′(a+) and f′(a), both exist and are finite. In other words f(x) is differentiable at x = a, if unique tangent can be drawn at this point.




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