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Successive Differentiation
  nᵗʰ Derivative of Function

Successive Differentiation

In general, the differentiation of a function can be differentiated succesively to obtain Higher Order Derivatives. Let function y=f(x) be a function of x. Then the result of differentiating y with respect to x is defined as the derivative or the first derivative of y with respect to x and is denoted as

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Then the result of differentiating ƒ(x) with respect to x is defined the second differential derivative of y with respect to x and is denoted as

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Similarly, the result of differentiating ƒ(x) with respect to x is defined the third differential derivative of y with respect to x and is denoted as 

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By continuing the differentiation process, the result of differentiating ƒ(x) n times with respect to x successively is defined the nth differential derivative of y with respect to x and is denoted as 

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The second, third, fourth, ..., nth derivatives are called higher order derivatives. 

nᵗʰ Derivative of Function

  1. nth Derivative of General Polynomial Function

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    Proof:

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  2. nth Derivative of General Natural Exponential Function

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    Proof:

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  3. nth Derivative of General Exponential Function

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    Proof:

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  4. nth Derivative of General Natural Logarithmic Function

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    Proof:

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  5. nth Derivative of General Sine Function

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    Proof:

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  6. nth Derivative of General Cosine Function

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    Proof:

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  7. nth Derivative of Product of General Functions

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    Proof:

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  8. nth Derivative of Product of General Functions

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    Proof:

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  9. nth Derivative of Product of Functions (Leibnitz's Theorem)

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    Proof:

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ID: 130700029 Last Updated: 7/15/2013 Revision: 0 Ref:

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References

  1. S. James, 1999, Calculus
  2. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering
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