Enter the expression you wish to differentiate below.
Operators: + , - , *, /, ^ . Constants: e , pi . Functions: sqrt , log (natural logarithm) , sin , cos , tan , sec , csc , cot
The derivative with respect to is:
12*x^2 + 6*x + 2
A derivative is a fundamental concept in calculus that describes how a function changes as its input changes. More precisely, it measures the rate of change or the slope of the function at a particular point. If you imagine a graph of a function, the derivative tells you how steep the curve is at any given point.
In everyday terms, think of driving a car. If your position on a road is represented as a function of time, the derivative of that function is your speed. It shows how your position changes every second. A higher derivative value means you're speeding up, while a negative value means you're moving in the opposite direction.
Mathematically, if we have a function f(x), the derivative of that function with respect to x is written as f'(x) or df/dx. It is defined as the limit:
f'(x) = lim(h → 0) [f(x + h) - f(x)] / h
This formula calculates the slope of the line that touches the function at exactly one point, known as the tangent line. The smaller the value of h, the closer we get to the exact slope at point x.
Derivatives are useful in a wide range of real-world applications. For example:
The Derivative Calculator is a simple and interactive tool designed to help you compute the derivative of mathematical expressions quickly and accurately. It’s especially useful for students, educators, engineers, and anyone learning or working with calculus.
This tool allows you to enter a mathematical expression and a variable (usually x) and instantly see the derivative of the expression with respect to that variable. It uses the powerful math.js library to handle the parsing and differentiation of the expression behind the scenes.
Whether you’re dealing with polynomials, trigonometric functions, logarithms, or exponential expressions, the calculator provides real-time results as you type, helping you understand the behavior and slope of functions without needing to solve derivatives by hand.
Here’s what you can expect from the Derivative Calculator:
The Derivative Calculator includes two main input fields that you need to use in order to get the correct result: the Expression Field and the Variable Field. Here's how each one works:
This is where you enter the mathematical expression that you want to differentiate. It supports a variety of operators, constants, and functions, making it versatile for many types of problems.
Examples of valid expressions:
4*x^3 + 3*x^2 + 2*x + 1sin(x) + cos(x)log(x) * x^2e^x + pi * x
Make sure to use the correct syntax, such as * for multiplication (e.g., write 2*x, not 2x) and ^ for powers (e.g., x^2).
In this field, you specify the variable with respect to which you want to differentiate the expression — most commonly x. However, you can also use other letters like y or z, as long as they are valid variable names.
Example:
3*y^2 + 2*y and you want the derivative with respect to y, enter y in the variable field.⚠️ Note: Variable names must start with a letter (A-Z or a-z). Using symbols or numbers as variable names will result in an error.
The Derivative Calculator includes two main input fields that you need to use in order to get the correct result: the Expression Field and the Variable Field. Here's how each one works:
This is where you enter the mathematical expression that you want to differentiate. It supports a variety of operators, constants, and functions, making it versatile for many types of problems.
Examples of valid expressions:
4*x^3 + 3*x^2 + 2*x + 1sin(x) + cos(x)log(x) * x^2e^x + pi * x
Make sure to use the correct syntax, such as * for multiplication (e.g., write 2*x, not 2x) and ^ for powers (e.g., x^2).
In this field, you specify the variable with respect to which you want to differentiate the expression — most commonly x. However, you can also use other letters like y or z, as long as they are valid variable names.
Example:
3*y^2 + 2*y and you want the derivative with respect to y, enter y in the variable field.⚠️ Note: Variable names must start with a letter (A-Z or a-z). Using symbols or numbers as variable names will result in an error.
By correctly filling both the expression and variable fields, the calculator will automatically compute and display the derivative in real time.
The Derivative Calculator supports a wide range of mathematical operations, constants, and functions, allowing you to work with many types of expressions. Here's a breakdown of what you can use:
These are the fundamental operators for building expressions:
+ — Addition- — Subtraction* — Multiplication (e.g., 3*x)/ — Division (e.g., 1/x)^ — Exponentiation (e.g., x^2 for "x squared")You can use the following built-in mathematical constants:
e — Euler’s number, approximately 2.718pi — The constant π, approximately 3.14159The calculator can also handle a variety of common mathematical functions. These include:
sqrt(x) — Square root of xlog(x) — Natural logarithm (base e) of xsin(x) — Sine of x (x in radians)cos(x) — Cosine of xtan(x) — Tangent of xsec(x) — Secant of xcsc(x) — Cosecant of xcot(x) — Cotangent of x
Make sure to use parentheses properly when writing functions. For example, write sin(x) rather than sinx.
By combining these operators, constants, and functions, you can create a wide variety of expressions to explore and differentiate using the calculator.
To ensure the Derivative Calculator works properly, it's important to enter your expressions using the correct format and syntax. A small mistake in how you write the expression can lead to errors or unexpected results. This section will help you understand how to write expressions correctly and what common mistakes to avoid.
Here are some key rules to follow when entering your expression:
2*x instead of 2x. The calculator needs the * to recognize multiplication.sin(x + 1) instead of sin x + 1.^ for exponents, such as x^2 for "x squared".sin, log, pi, and e in lowercase letters.log(x) is correct; log x is not.3x instead of 3*x.sin x instead of sin(x).( must have a closing ). Example: (x + 1 is invalid.Sin(x) instead of sin(x) will not work.@, $, or % which are not valid in mathematical expressions.Taking a moment to double-check your input before calculating can save time and help you get accurate results.
While using the Derivative Calculator, you might occasionally see error messages. These messages are meant to guide you in correcting your input so the calculator can perform the differentiation correctly. Below are the most common error messages and what they mean:
This error appears when the variable you've entered is not valid. The calculator expects a variable name that starts with a letter (A–Z or a–z), such as x, y, or z.
Causes:
2, @, $).xy instead of just x.How to fix it: Use a single letter, like x, in the variable field.
This error indicates that the expression you entered cannot be understood or parsed by the calculator.
Causes:
3x instead of 3*x).sin x instead of sin(x)).(x+1 or x+1)).How to fix it:
( has a matching closing parenthesis ).Understanding these messages will help you quickly fix issues and get the correct derivative without confusion.
Once you enter a valid mathematical expression and a variable, the Derivative Calculator will display the derivative of your input with respect to the variable you chose. But what does this result mean, and how can you interpret it correctly?
The output shown by the calculator is the first derivative of the function you entered. This new expression represents the rate of change, or how fast the original function increases or decreases at any given point along the variable's axis.
For example, if you input the function f(x) = 4*x^3 + 3*x^2 + 2*x + 1, the calculator shows:
12*x^2 + 6*x + 2
This means that the slope of the original curve at any point x is given by this new expression. If you plug in a specific value of x into the derivative, you will get the slope of the tangent line to the curve at that point.
For example, if the derivative is 2*x and you plug in x = 3, the result is 6. This means that at x = 3, the slope of the original function is 6 — it’s going up steeply.
One of the most convenient features of the Derivative Calculator is its ability to provide real-time updates. As soon as you start typing or modifying your expression or variable, the calculator instantly recalculates and displays the updated derivative — no need to click a button or reload the page.
Behind the scenes, the calculator uses JavaScript and the math.js library to detect changes in the input fields. Each time you make a change to the expression or the variable, an event is triggered that tells the script to:
This process happens almost instantly, so you see the results as you type. It’s especially helpful when learning, because you can experiment with different expressions and immediately see how the derivative changes.
Real-time updates make the Derivative Calculator a powerful and user-friendly tool for both beginners and experienced users looking to explore calculus concepts efficiently.
Let’s walk through a clear example to see how the Derivative Calculator works and understand how it arrives at the correct derivative.
We’ll use the polynomial function:
4*x^3 + 3*x^2 + 2*x + 1
This is a basic cubic function where each term has a power of x. We want to find the derivative with respect to x.
To differentiate the expression, we apply the power rule to each term:
4*x^3 → Bring down the exponent (3) and subtract 1 from it:
4 * 3 * x^2 = 12*x^23*x^2 → 3 * 2 * x^1 = 6*x2*x → 2 * 1 * x^0 = 21 → Constant terms always have a derivative of 0Putting it all together, the derivative of the original expression is:
12*x^2 + 6*x + 2
The output 12*x^2 + 6*x + 2 is the derivative of the input function. It tells you how fast the original function is changing at any point along the x-axis.
For example:
x = 1, the slope is 12*1^2 + 6*1 + 2 = 20x = 0, the slope is 12*0 + 6*0 + 2 = 2x = -1, the slope is 12*1 - 6 + 2 = 8
This means the original function is increasing faster as x becomes larger. Understanding this behavior helps in graphing, optimization, and analyzing trends in real-world applications.
The Derivative Calculator isn't just for simple expressions — it's also powerful enough to handle more complex calculus problems. Here are some useful tips for advanced users who want to get the most out of the tool.
You can enter expressions with functions inside of other functions, commonly known as nested functions. The calculator automatically applies the chain rule to differentiate them.
Examples:
sin(x^2) → The calculator applies the chain rule to give 2*x*cos(x^2).log(sqrt(x)) → This is interpreted and differentiated as 1 / (2*x).e^(3*x + 1) → The derivative is 3 * e^(3*x + 1).
💡 Tip: Always use parentheses to clearly define the structure of nested functions, like cos(x^2) or log(x+1).
The calculator can handle standard trigonometric functions including sin, cos, tan, sec, csc, and cot. It also correctly applies derivative rules to combinations of these functions.
Examples:
sin(x) → cos(x)cos(x) → -sin(x)tan(x) → sec(x)^2sec(x) → sec(x) * tan(x)You can also combine trigonometric functions with powers or nested expressions:
sin(x^2) → 2*x*cos(x^2)x*sin(x) → x*cos(x) + sin(x) (using the product rule)(x^2 + 1)^3 to test chain rule applications.(x^2 + 1)/(x + 2).e^x + log(x) + sin(x).With the right formatting and understanding of how functions behave, you can explore a wide range of calculus problems and instantly get accurate results.
x?
Yes! You can use any single letter as a variable (like y or z). Just make sure it starts with a letter and matches what's used in your expression.
This version of the calculator only supports first derivatives. For second or higher-order derivatives, you'll need to take the output and differentiate it again manually or using another tool.
If the expression is invalid (e.g., missing operators or unmatched parentheses), the calculator will display an error message. Make sure your input follows proper syntax and includes all necessary operators.
No, the calculator currently does not support piecewise functions or expressions with absolute values like |x|. It is designed for continuous and differentiable expressions.
Yes, it is. Use lowercase letters for functions like sin, log, sqrt, and constants like e and pi. Writing SIN(x) or LOG(x) will not work.
Absolutely! You can highlight the output with your mouse and copy it like normal text to use it elsewhere, such as in homework, documents, or further calculations.
Use the caret symbol ^ for exponents (e.g., x^2) and the asterisk * for multiplication (e.g., 3*x). Implicit multiplication like 3x is not allowed.
Yes, you can use both decimals (e.g., 0.5*x) and fractions (e.g., 1/2*x) in your expressions.