Derivative Calculator

- May 09, 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))

What is a Derivative Calculator?

A derivative calculator is a handy tool that helps you find the derivative of mathematical functions. This means it shows how a function changes when its input changes. With this calculator, you can work with various types of functions, including polynomials, trigonometric functions, logarithmic, and exponential expressions. It’s an essential tool for anyone studying Calculus or needing to analyse the behaviour of functions.

Easy Function Input

Using the derivative calculator is simple. You start by entering a function into the input box. For example, you might type something like (x^2 + \sin(x)). After that, you can select the order of the derivative you wish to calculate, whether it's the first, second, or even up to the fifth derivative. This flexibility makes the calculator suitable for a broad range of mathematical problems.

Customising Your Calculations

The calculator offers customisation options that enhance your experience. You can choose the variable you want to differentiate, such as (x) or (y). Additionally, there’s an option to evaluate the derivative at a specific point. If you want to know the slope of the function at (x=1), for instance, you can input that directly and receive immediate feedback on your calculations.

Display Options to Suit Your Needs

When using the derivative calculator, you can adjust how results are presented. You can select different output formats, such as simplified results, detailed steps, or even LaTeX for more formal presentations. Additionally, there are options to display differentiation rules and graphical representations of the function and its derivative. This way, you can see not only the answer but also how it was derived.

Understanding Derivatives and their Importance

Derivatives are a core concept in calculus. They help you understand the rate of change of a function's output relative to its input. This understanding is crucial in various fields, including Physics, engineering, and economics. Knowing how to calculate derivatives allows you to analyse trends, optimise processes, and solve real-world problems effectively.

Common Differentiation Rules

When calculating derivatives, certain rules are frequently used. Here are a few key rules:

  • Power Rule: If \(f(x) = x^n\), then \(f'(x) = n \cdot x^{n-1}\)
  • Product Rule: If \(h(x) = f(x) \cdot g(x)\), then \(h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)\)
  • Quotient Rule: If \(h(x) = \frac{f(x)}{g(x)}\), then \(h'(x) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\)

These rules simplify the process of finding derivatives and are essential for anyone working with calculus.

Trigonometric and Other Functions

Different functions have specific derivatives, especially trigonometric, exponential, and logarithmic functions. For example:

  • \(d/dx[\sin(x)] = \cos(x)\)
  • \(d/dx[e^x] = e^x\)
  • \(d/dx[\ln(x)] = \frac{1}{x}\)

Knowing these key derivatives helps you to calculate derivatives faster and more accurately.

Getting Started with the Derivative Calculator

To sum up, the derivative calculator is an effective tool for anyone looking to understand or compute derivatives. Whether you’re a student or a professional, this calculator can help you tackle complex mathematical problems. With its user-friendly interface and customisation options, it’s designed to make the process of calculating derivatives both straightforward and informative.