Calculator for Derivatives

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Expert Reviewed: This Derivative Calculator and its underlying financial formulas have been verified for accuracy by David Chen, CFA.

This calculator uses the highly accurate Central Difference Method to numerically approximate the derivative $f'(a)$ of any given function $f(x)$ at a specific point $x=a$. Enter your function and parameters below to get the instantaneous rate of change.

Derivative Calculator

Derivative Calculator Formula

This calculator uses the Central Difference Quotient formula:

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

Formula Source: MIT OpenCourseWare – Single Variable Calculus | Khan Academy – Limit Definition

Variables Explained

  • Function $f(x)$: The mathematical expression you want to differentiate (e.g., $x^2$, $\sin(x)$).
  • Point $a$: The specific point on the x-axis where you want to find the instantaneous rate of change.
  • Step Size $h$: A very small positive number ($\Delta x$) used for the approximation. A smaller $h$ usually means higher accuracy, but too small can cause floating-point errors.

What is a Derivative Calculator?

A derivative calculator is an essential tool in calculus that determines the rate at which a function’s value changes with respect to a change in its independent variable. In simple terms, it calculates the slope of the tangent line to the function at a specific point. This slope represents the instantaneous rate of change.

The concept of the derivative is fundamental in physics, economics, and engineering. For instance, in physics, the derivative of a position function gives the instantaneous velocity, and the derivative of the velocity function gives the instantaneous acceleration. In finance, it can be used to model rates of change in investment returns or pricing.

How to Calculate a Derivative (Example)

Let’s find the derivative of $f(x) = x^3 + 2x$ at the point $a=2$ using the step size $h=0.000001$.

  1. Define the Function and Parameters: $f(x) = x^3 + 2x$, $a=2$, and $h=0.000001$.
  2. Calculate $f(a+h)$: Calculate $f(2 + 0.000001) = f(2.000001)$.
  3. Calculate $f(a-h)$: Calculate $f(2 – 0.000001) = f(1.999999)$.
  4. Apply the Central Difference Formula: $$f'(2) \approx \frac{f(2.000001) – f(1.999999)}{2 \times 0.000001}$$
  5. Result: The exact derivative is $f'(2) = 14$. The numerical approximation should be extremely close to 14.000000.

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Frequently Asked Questions (FAQ)

Why do we use the Central Difference Method?

It provides a significantly more accurate approximation of the derivative compared to the basic forward or backward difference methods, especially when the step size $h$ is very small.

What is the derivative?

The derivative measures how sensitive a function’s output is to changes in its input. Geometrically, it’s the slope of the tangent line to the curve at a specific point.

What kind of functions can I enter?

You can enter any valid mathematical expression using standard JavaScript syntax (e.g., Math.pow(x, 2) for $x^2$, Math.sin(x) for $\sin(x)$). You must use ‘x’ as the variable name.

What happens if the step size ‘h’ is too small?

If $h$ is too small (e.g., less than $10^{-15}$), the subtraction in the numerator ($f(a+h) – f(a-h)$) can lead to catastrophic cancellation due to floating-point precision limits, reducing the accuracy of the result.

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