# Central Difference Formula | Example, First & Second Derivative

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135 # Central Difference Formula | Example, First, Second Derivative

The central difference formula is a method for approximating the derivative of a function at a particular point. It is based on the idea of using the values of the function at nearby points to estimate the slope of the function at the point of interest.

The central difference formula is given by:

f'(x) ≈ [f(x+h) – f(x-h)] / (2*h)

where h is a small positive number called the step size.

The central difference formula is an example of a finite difference formula, which is a method for approximating derivatives using the values of the function at a finite number of points.

The central difference formula has the following properties:

• It is an accurate approximation of the derivative when h is small.
• It is an unbiased estimator of the derivative, meaning that it has an expected error of zero.
• It has a smaller error than the forward or backward difference formulas when h is small.

• Finite difference formulas are used to approximate the derivative of a function at a particular point using the values of the function at a finite number of points. There are many different finite difference formulas that can be used, depending on the desired level of accuracy and the number of function values that are available.
• The central difference formula is an example of a second-order finite difference formula, meaning that it uses the values of the function at three points (x-h, x, and x+h) to estimate the derivative. Higher-order finite difference formulas use the values of the function at more points to obtain more accurate approximations of the derivative.
• Finite difference formulas can be used to approximate the derivative of a function when the derivative is not known explicitly or when it is difficult to calculate directly. They are commonly used in numerical analysis, scientific computing, and other fields where accurate approximations of derivatives are needed.
• The error in a finite difference formula depends on the step size h and the smoothness of the function. For a given function, the error decreases as the step size decreases, but the cost of calculating the derivative increases. Therefore, it is important to choose an appropriate step size that balances accuracy and efficiency.

## Central Difference Formula | Example

Here is an example of how to use the central difference formula to approximate the derivative of a function:

Example:

Suppose we want to approximate the derivative of the function f(x) = x^2 at x = 1. We can use the central difference formula as follows:

f'(1) ≈ [f(1+h) – f(1-h)] / (2*h)

Let’s take h = 0.1. Then:

f'(1) ≈ [f(1.1) – f(0.9)] / (20.1) ≈ [(1.1)^2 – (0.9)^2] / (20.1) ≈ (1.21 – 0.81) / 0.2 ≈ 2

Therefore, the approximate derivative of f(x) at x = 1 is 2.

## Central difference formula for approximating the first derivative of a function:

f'(x) ≈ [f(x+h) – f(x-h)] / (2*h)

This formula can be used to approximate the derivative of a function at a particular point x by using the values of the function at the nearby points x+h and x-h.

The step size h is a small positive number that determines the accuracy of the approximation. The smaller the value of h, the more accurate the approximation will be, but the cost of calculating the derivative will also increase.

Here is an example of how to use the central difference formula to approximate the first derivative of a function:

Example:

Suppose we want to approximate the first derivative of the function f(x) = x^2 at x = 1. We can use the central difference formula as follows:

f'(1) ≈ [f(1+h) – f(1-h)] / (2*h)

Let’s take h = 0.1. Then:

f'(1) ≈ [f(1.1) – f(0.9)] / (20.1) ≈ [(1.1)^2 – (0.9)^2] / (20.1) ≈ (1.21 – 0.81) / 0.2 ≈ 2

Therefore, the approximate first derivative of f(x) at x = 1 is 2.

## Central difference formula for approximating the second derivative of a function:

f”(x) ≈ [f(x+h) – 2*f(x) + f(x-h)] / (h^2)

This formula can be used to approximate the second derivative of a function at a particular point x by using the values of the function at the nearby points x+h and x-h.

The step size h is a small positive number that determines the accuracy of the approximation. The smaller the value of h, the more accurate the approximation will be, but the cost of calculating the derivative will also increase.

Here is an example of how to use the central difference formula to approximate the second derivative of a function:

Example:

Suppose we want to approximate the second derivative of the function f(x) = x^2 at x = 1. We can use the central difference formula as follows:

f”(1) ≈ [f(1+h) – 2*f(1) + f(1-h)] / (h^2)

Let’s take h = 0.1. Then:

f”(1) ≈ [f(1.1) – 2f(1) + f(0.9)] / (0.1^2) ≈ [(1.1)^2 – 2(1)^2 + (0.9)^2] / (0.1^2) ≈ (1.21 – 1.8 + 0.81) / 0.01 ≈ 0

Therefore, the approximate second derivative of f(x) at x = 1 is 0.