This document discusses numerical differentiation, which is the process of finding the numerical value of the derivative of a given function at a given point. It notes that if the x-values are equally spaced, derivatives are calculated using Newton's forward or backward interpolation formulas, while if they are not equally spaced, a difference formula is used. Examples are provided to demonstrate calculating derivatives from given data using finite difference methods like Newton's forward and backward difference formulas.

1. NUMERICAL ANALYSIS (15UMTC64)
III B.SC. MATHEMATICS (SF)
Ms. M. Mohanamalar, M.Sc., M.Phil.,
Ms. N. Malathi, M.Sc.,

2. • Calculus is the mathematics of change. Because engineers
must continuously deal with systems and processes that
change, calculus is an essential tool of engineering.
• Standing in the heart of calculus are the mathematical concepts
of differentiation and integration
• Numerical differentiation is the process of finding the
numerical value of the derivative of a given function at a given
point.

3. For some particular value of x from the given data (xi,yi), i=1, 2,
3…..n where y=f(x) i explicitly, the interpolation to be used depends on
the particular value of x which derivatives are required.
•If the values of x are not equally spaced, we represent the function by
difference formula and the derivatives are obtained.
•If the values of x are equally spaced, the derivatives are calculated by
using Newton’s Forward interpolation formula or backward interpolation
formula.

4. Principle:
First fit a polynomial for the given difference data interpolation using
Newton ’s divided difference interpolation formula and compute the
derivatives for a given x.

5. Example

6. Example

7. erivatives using Finite difference
Newton’s Forward Difference formula is
Newton’s Backward Difference formula is

8. erivatives using Finite difference

9. erivatives using Finite difference

10. erivatives using Finite difference
Example 2.
Find the first two derivatives of y at x=54 from the following data
x 50 51 52 53 54
y 3.6840 3.7083 3.7325 3.7563 3.7798

11. erivatives using Finite difference
Example

12. erivatives using Finite difference
Example 1
Find first and second derivatives of the function at the point x=1.2
from the following data
x 1 2 3 4 5
y 0 1 5 6 8
The difference table

13. erivatives using Finite difference
Example