1. Numerical Differentiation
When function is given as
Analytical expression
Values at discrete points
For complicated functions and discrete pt values;
Derivatives replaced by discrete forms
Differentiation done by finite difference approximation

2. Engineering Applications
Most engineering and scientific problems involve derivatives
Temperature distribution
Response of mass-spring to excitation (gun recoil, shock absorbers, etc.)
Many others..

3. Definition of Derivatives
For finite differences;
𝑑𝑓𝑥 𝑑𝑥 𝑥 0 =𝑓′ 𝑥 0 = lim 𝑥 𝑥 0 𝑓𝑥−𝑓 𝑥 0  𝑥− 𝑥 0
𝑓′ 𝑥 0 ≈ 𝑓𝑥−𝑓 𝑥 0  𝑥

4. Basic Finite Diff. Approximations
Two point forward difference
Two point backward difference
Two point central difference
𝑑𝑓 𝑑𝑥 𝑖 ≈ 𝑓 𝑥 = 𝑓 𝑖1 − 𝑓 𝑖 𝑥 𝑖1 − 𝑥 𝑖
𝑑𝑓 𝑑𝑥 𝑖 ≈ 𝑓 𝑥 = 𝑓 𝑖 − 𝑓 𝑖−1 𝑥 𝑖 − 𝑥 𝑖−1
𝑑𝑓 𝑑𝑥 𝑖 ≈ 𝑓 𝑥 = 𝑓 𝑖1 − 𝑓 𝑖−1 𝑥 𝑖1 − 𝑥 𝑖−1

5. Based on Taylor Series
𝑓 𝑥 𝑖1 ≃𝑓 𝑥 𝑖 𝑓′ 𝑥 𝑖  𝑥 𝑖1 − 𝑥 𝑖  𝑓′ 𝑥 𝑖  2!  𝑥 𝑖1 − 𝑥 𝑖  2  𝑓′′ 𝑥 𝑖  3!  𝑥 𝑖1 − 𝑥 𝑖  3  𝑓 𝑛  𝑥 𝑖  𝑛!  𝑥 𝑖1 − 𝑥 𝑖  𝑛  𝑅 𝑛

6. Finite divided difference
Truncating 2nd order & higher terms & rearrange
First forward difference
𝑓′ 𝑥 𝑖 = 𝑓 𝑥 𝑖1 −𝑓 𝑥 𝑖  𝑥 𝑖1 − 𝑥 𝑖 𝑂 𝑥 𝑖1 − 𝑥 𝑖  = 𝑓 𝑥 𝑖1 −𝑓 𝑥 𝑖  ℎ =  𝑓 𝑖 ℎ 𝑂ℎ