UNIT 3 • NUMERICAL DIFFERENTIATION

Numerical
Differentiation Method

When symbolic calculus is impossible or discrete experimental data is collected, Numerical Differentiation uses symmetric finite difference stencils to compute high-precision derivative slopes $f'(x)$ and curvatures $f''(x)$ !

CENTRAL DIFFERENCE STENCILSO(h²) ACCURACY

First Derivative (Slope)

f'(x) \approx \frac{f(x + h) - f(x - h)}{2h}

Second Derivative (Curvature)

f''(x) \approx \frac{f(x + h) - 2f(x) + f(x - h)}{h^2}

1. Target Point (x)The exact evaluation point where rates of change are measured.

2. Small Step Size (h)Symmetric perturbations

x + h

and

x - h

3. Error CancellationSymmetry cancels odd-order Taylor error terms automatically.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Consider the cubic polynomial function:

f(x) = x^3 - 2x + 5

Let's evaluate $f'(2)$ and $f''(2)$ using step size $h = 0.1$ !

STEP 01 • EVALUATE FUNCTION AT NEIGHBORING POINTS

Compute $f(x-h), f(x),$ and $f(x+h)$ around $x = 2$ :

f(1.9) = (1.9)^3 - 2(1.9) + 5 = 6.859 - 3.8 + 5 = 8.059

f(2.0) = (2.0)^3 - 2(2.0) + 5 = 8 - 4 + 5 = 9.000

f(2.1) = (2.1)^3 - 2(2.1) + 5 = 9.261 - 4.2 + 5 = 10.061

STEP 02 • COMPUTE FIRST DERIVATIVE f'(2)

Plug functional values into central first derivative formula:

f'(2) \approx \frac{10.061 - 8.059}{2(0.1)} = \frac{2.002}{0.2} = 10.0100

(Exact calculus value: $f'(x) = 3x^2 - 2 \implies f'(2) = 10$ ).

STEP 03 • COMPUTE SECOND DERIVATIVE f''(2)

Plug values into central second derivative curvature formula:

f''(2) \approx \frac{10.061 - 2(9.000) + 8.059}{0.1^2} = \frac{0.120}{0.01} = 12.0000

(Exact calculus value: $f''(x) = 6x \implies f''(2) = 12$ ).

FINAL RESULT

Estimated Derivatives: f'(2) = 10.0100 , f''(2) = 12.0000

\boxed{f'(2) \approx 10.0100, \quad f''(2) \approx 12.0000}

With $h = 0.1$ , central numerical differentiation matches the exact analytical slope 10 and curvature 12 to extraordinary accuracy!

LABORATORY

Interactive Live Calculator & Visualizer

FINITE DIFFERENCE DERIVATIVE LABORATORY

Numerical Differentiation Engine

PreviousWeddle's Rule

Unit 3•17 of 28

NextTaylor's Series Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 3 • NUMERICAL DIFFERENTIATION

Numerical
Differentiation Method

CENTRAL DIFFERENCE STENCILSO(h²) ACCURACY

First Derivative (Slope)

f'(x) \approx \frac{f(x + h) - f(x - h)}{2h}

Second Derivative (Curvature)

f''(x) \approx \frac{f(x + h) - 2f(x) + f(x - h)}{h^2}

1. Target Point (x)The exact evaluation point where rates of change are measured.

2. Small Step Size (h)Symmetric perturbations

x + h

and

x - h

3. Error CancellationSymmetry cancels odd-order Taylor error terms automatically.

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Consider the cubic polynomial function:

f(x) = x^3 - 2x + 5

Let's evaluate $f'(2)$ and $f''(2)$ using step size $h = 0.1$ !

STEP 01 • EVALUATE FUNCTION AT NEIGHBORING POINTS

Compute $f(x-h), f(x),$ and $f(x+h)$ around $x = 2$ :

f(1.9) = (1.9)^3 - 2(1.9) + 5 = 6.859 - 3.8 + 5 = 8.059

f(2.0) = (2.0)^3 - 2(2.0) + 5 = 8 - 4 + 5 = 9.000

f(2.1) = (2.1)^3 - 2(2.1) + 5 = 9.261 - 4.2 + 5 = 10.061

STEP 02 • COMPUTE FIRST DERIVATIVE f'(2)

Plug functional values into central first derivative formula:

f'(2) \approx \frac{10.061 - 8.059}{2(0.1)} = \frac{2.002}{0.2} = 10.0100

(Exact calculus value: $f'(x) = 3x^2 - 2 \implies f'(2) = 10$ ).

STEP 03 • COMPUTE SECOND DERIVATIVE f''(2)

Plug values into central second derivative curvature formula:

f''(2) \approx \frac{10.061 - 2(9.000) + 8.059}{0.1^2} = \frac{0.120}{0.01} = 12.0000

(Exact calculus value: $f''(x) = 6x \implies f''(2) = 12$ ).

FINAL RESULT

Estimated Derivatives: f'(2) = 10.0100 , f''(2) = 12.0000

\boxed{f'(2) \approx 10.0100, \quad f''(2) \approx 12.0000}

With $h = 0.1$ , central numerical differentiation matches the exact analytical slope 10 and curvature 12 to extraordinary accuracy!

LABORATORY

Interactive Live Calculator & Visualizer

FINITE DIFFERENCE DERIVATIVE LABORATORY

Numerical Differentiation Engine

PreviousWeddle's Rule

Unit 3•17 of 28

NextTaylor's Series Method

Numerical Differentiation Method

Step-by-Step Worked Example

Estimated Derivatives: f'(2) = 10.0100 , f''(2) = 12.0000

Interactive Live Calculator & Visualizer

Numerical Differentiation Engine

NETZ Engine

Numerical Differentiation Method

Step-by-Step Worked Example

Estimated Derivatives: f'(2) = 10.0100 , f''(2) = 12.0000

Interactive Live Calculator & Visualizer

Numerical Differentiation Engine

Numerical
Differentiation Method

Numerical
Differentiation Method