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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

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

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

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

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

nᵗʰ Derivative of Function

1. nth Derivative of General Polynomial Function

Proof:

2. nth Derivative of General Natural Exponential Function

Proof:

3. nth Derivative of General Exponential Function

Proof:

4. nth Derivative of General Natural Logarithmic Function

Proof:

5. nth Derivative of General Sine Function

Proof:

6. nth Derivative of General Cosine Function

Proof:

7. nth Derivative of Product of General Functions

Proof:

8. nth Derivative of Product of General Functions

Proof:

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

Proof:

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

References

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

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