Derivative Calculator

Category: Calculus
- May 14, 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 handy tool designed to help users compute the derivative of various mathematical functions. It accommodates a range of functions, including polynomials, trigonometric functions, logarithms, and exponential expressions. With just a few inputs, anyone can find the rate of change of a function at any given point, making Calculus more accessible to students and enthusiasts alike.

How to Enter Your Function

To begin, simply input the function you want to differentiate in the designated field. You could enter functions like x^2 + sin(x) or e^x. The calculator will parse the input and prepare for differentiation. It’s crucial to ensure that the function is entered correctly to receive accurate results.

Selecting the Derivative Order

The calculator allows users to choose the order of the derivative they wish to calculate. Options typically range from the first derivative to the fifth derivative. The first derivative shows the immediate rate of change, while higher-order derivatives can provide insights into the function's behaviour over time. Understanding the different orders of derivatives can be vital for more complex analyses.

Evaluation at a Specific Point

One of the special features of this calculator is the option to evaluate the derivative at a specific point. If you check this option, you will be prompted to enter a number corresponding to the value of x. This allows you to see how the function behaves at that particular point, which can be particularly useful in practical applications.

Customising Output Options

The Derivative Calculator offers several output options to tailor the results to your needs. You can select from formats such as simplified results, detailed steps, or LaTeX format. This flexibility ensures that whether you need a straightforward answer or a comprehensive breakdown of the differentiation process, you get what you require.

Visualising Derivatives with Graphs

In addition to textual results, the calculator provides a graphical representation of the function and its derivative. This feature helps users visually understand the relationship between the function and its rate of change. The graph clearly distinguishes between the original function and its derivative, making it easier to grasp complex concepts.

Common Differentiation Rules

When using the Derivative Calculator, it's helpful to know some common differentiation rules, which include:

  • Power Rule: If f(x) = xn, then f'(x) = n·xn-1
  • 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

Why Use the Derivative Calculator?

The Derivative Calculator is not just a tool for students but also for anyone involved in fields that require calculus. Whether you’re studying for an exam, solving real-world problems, or simply exploring mathematical concepts, this calculator makes it easier to perform complex calculations. It simplifies the process, allowing you to focus on understanding the underlying concepts rather than getting bogged down in tedious computations.