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FUNDAMENTALS OF MATHEMATICS DIFFERENTIAL CALCULUS by Sanjay Mishra

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ALGEBRA OF DIFFERENTIABILITY

If f(x) and g(x) are differentiable functions at x = a, then following holds good

  1. kf(x) is always differentiable ( k is finite) at x = a

    Proof: Since f(x) is differentiable at x = a

    images finite and real

    images

    kf is differentiable at x = a

     

  2. f(x) ± g(x) is always differentiable at x = a

    Proof: Let ψ(x) = f(x) + g(x)

    images

    = f'(a+) + g'(a+) = finite

    f'(a) = f'(a+) and g'(a) = g'(a+)

    ∴ L.H.D = R.H.D at x = a for ψ(x)

    Thus ( ...

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