Derivative Calculator

Category: Calculus
- April 01, 2025
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Calculate the derivative of mathematical functions. This calculator supports common functions including polynomials, trigonometric, logarithmic, and exponential expressions.

Function Input

Evaluation (Optional)

Display Options

About Derivatives

In calculus, the derivative measures how a function changes as its input changes. Specifically, the derivative of a function at a point measures the rate at which the function's output changes per unit change in its input.

Common Differentiation Rules
  • Power Rule: If f(x) = xn, then f'(x) = n·xn-1
  • Sum Rule: If h(x) = f(x) + g(x), then h'(x) = f'(x) + g'(x)
  • Product Rule: If h(x) = f(x)·g(x), then h'(x) = f'(x)·g(x) + f(x)·g'(x)
  • Quotient Rule: If h(x) = f(x)/g(x), then h'(x) = [f'(x)·g(x) - f(x)·g'(x)]/[g(x)]2
  • Chain Rule: If h(x) = f(g(x)), then h'(x) = f'(g(x))·g'(x)
Trigonometric Functions
  • d/dx[sin(x)] = cos(x)
  • d/dx[cos(x)] = -sin(x)
  • d/dx[tan(x)] = sec2(x)
  • d/dx[sec(x)] = sec(x)·tan(x)
  • d/dx[csc(x)] = -csc(x)·cot(x)
  • d/dx[cot(x)] = -csc2(x)
Exponential and Logarithmic Functions
  • d/dx[ex] = ex
  • d/dx[ax] = ax·ln(a)
  • d/dx[ln(x)] = 1/x
  • d/dx[loga(x)] = 1/(x·ln(a))

Understanding the Derivative Calculator

The Derivative Calculator is a practical tool designed to compute the derivative of a given function. Whether you're studying calculus or solving real-world problems involving rates of change, this tool makes calculating derivatives simple, while providing step-by-step solutions and visualizations.

What Are Derivatives?

A derivative measures how a function changes as its input changes. It represents the slope of the function's graph at any given point. In simple terms, derivatives help answer questions like: - How fast is something changing at a specific moment? - What is the rate of increase or decrease in a particular situation?

For example: - In physics, the derivative of a position function gives velocity. - In business, the derivative of a cost function can show the marginal cost.

Mathematically, if ( f(x) ) is a function, then its derivative ( f'(x) ) is given by: f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h

Key Features of the Calculator

  • Accurate Derivative Calculation:

  • Compute derivatives for a variety of common mathematical functions with ease.

  • Step-by-Step Explanations:

  • Understand the process of differentiation with detailed steps.

  • Graph Visualization:

  • Compare the input function and its derivative on an interactive graph.

  • Preloaded Examples:

  • Experiment with preloaded examples like x^3 + sin(x), e^x + x^2, and more.

  • Mobile-Friendly Design:

  • Works seamlessly on both desktop and mobile devices.

How to Use the Derivative Calculator

  1. Input a Function:

  2. Type your function into the input field labeled Enter a function. For example, you could enter x^3 + sin(x).

  3. Select an Example (Optional):

  4. Use the dropdown menu to choose from preloaded examples like e^x + x^2. The input field will automatically update with the selected example.

  5. Click Calculate:

  6. Press the Calculate button to generate results, including:

    • The derivative in standard mathematical notation.
    • A step-by-step breakdown of the calculation.
    • A graph displaying both the original function and its derivative.

  7. Clear the Input:

  8. Press the Clear button to reset the calculator and start over.

Example Walkthroughs

Example 1: x^3 + sin(x)

  • Derivative: 3x^2 + cos(x)
  • Steps:
  • The derivative of x^3 is 3x^2.
  • The derivative of sin(x) is cos(x).
  • Combine the results: 3x^2 + cos(x).
  • Graph: The graph displays the input function x^3 + sin(x) alongside its derivative 3x^2 + cos(x).

Example 2: e^x + x^2

  • Derivative: e^x + 2x
  • Steps:
  • The derivative of e^x is e^x.
  • The derivative of x^2 is 2x.
  • Combine the results: e^x + 2x.
  • Graph: The graph shows the input function e^x + x^2 and its derivative e^x + 2x.

Example 3: ln(x)

  • Derivative: 1 / x
  • Steps:
  • The derivative of ln(x) is 1 / x.
  • Graph: The graph illustrates the natural logarithmic function ln(x) and its derivative 1 / x.

Example 4: x^2 * sin(x)

  • Derivative: 2x * sin(x) + x^2 * cos(x)
  • Steps:
  • Use the product rule for differentiation.
  • Differentiate x^2 as 2x.
  • Differentiate sin(x) as cos(x).
  • Combine the results: 2x * sin(x) + x^2 * cos(x).
  • Graph: The graph compares the input function x^2 * sin(x) and its derivative 2x * sin(x) + x^2 * cos(x).

Why Use This Calculator?

This Derivative Calculator simplifies differentiation for everyone, from students learning calculus to professionals solving practical problems. Here's why it’s useful:

  • Educational Tool:

  • Gain a better understanding of differentiation through step-by-step solutions.

  • Graphical Representation:

  • Visualize the relationship between a function and its derivative.

  • Ease of Use:

  • Quickly calculate derivatives without manual computations.

Try It Now

Explore how derivatives reveal the rate of change in various scenarios. Input your function, calculate, and see the results with detailed steps and visualizations. Get started today!