Updated on April 17th, 2026

Derivative Calculator

Created By Jehan Wadia

DERIVATIVE RESULT
Step-by-Step Solution
Interactive Graph
Practice Mode

Find the derivative of:

Common Derivatives Reference
Function f(x) Derivative f'(x) Rule

Introduction

A derivative tells you how fast something is changing at any given point. It is one of the most important ideas in calculus. When you take the derivative of a function, you find its rate of change — like how speed tells you how fast your position changes over time. This Derivative Calculator lets you type in any function and get the derivative right away. It handles common rules like the power rule, product rule, quotient rule, and chain rule so you don't have to work them out by hand. Whether you are learning calculus for the first time or just need a quick answer to check your work, this tool makes finding derivatives simple and fast.

How to Use Our Derivative Calculator

Enter a math function and this calculator will find its derivative for you step by step.

Function f(x): Type the function you want to differentiate. Use "x" as your variable. For example, you can enter expressions like x^2, sin(x), 3x+5, or ln(x). You can also use parentheses to group terms, such as (x+1)^3.

Order of Derivative: Choose which derivative you want to find. Pick 1 for the first derivative, 2 for the second derivative, and so on. The first derivative tells you the slope or rate of change of your function. Higher-order derivatives give you the derivative of the derivative.

Point to Evaluate (optional): If you want to know the value of the derivative at a specific x-value, enter that number here. For instance, if you enter 3, the calculator will plug in x = 3 into the derivative and give you a numerical answer.

Once you have filled in your inputs, click the calculate button. The calculator will display the derivative expression, show the simplified result, and provide the evaluated value if you entered a specific point.

A derivative measures how a function changes as its input changes. In simple terms, it tells you the rate of change or the slope of a function at any given point. If you have a function that describes the position of a car over time, the derivative of that function gives you the car's speed. Derivatives are one of the two main ideas in calculus, and they show up everywhere in math, science, engineering, and economics.

How Derivatives Work

The derivative of a function f(x) is written as f'(x) or df/dx. It is found by looking at how much the output of the function changes when you make a tiny change to the input. For example, if f(x) = x², the derivative is f'(x) = 2x. This means that at x = 3, the slope of the curve is 6, and at x = 5, the slope is 10. The function gets steeper as x gets larger.

Key Derivative Rules

There are several important rules that make finding derivatives easier:

  • Power Rule: The derivative of xn is n·xn−1. For example, the derivative of x³ is 3x².
  • Constant Rule: The derivative of any constant number (like 5 or −3) is 0, because constants don't change.
  • Sum/Difference Rule: The derivative of a sum or difference is just the sum or difference of the individual derivatives. So d/dx(x² + 3x) = 2x + 3.
  • Product Rule: When two functions are multiplied together, the derivative is f'g + fg'. You differentiate one part at a time while keeping the other part the same.
  • Quotient Rule: For a fraction f/g, the derivative is (f'g − fg') / g².
  • Chain Rule: When a function is inside another function, like sin(x²), you multiply the outer derivative by the inner derivative. This is one of the most used rules in calculus.

Common Derivatives to Know

Some derivatives come up so often that they're worth memorizing. The derivative of sin(x) is cos(x). The derivative of cos(x) is −sin(x). The derivative of ex is ex (it's its own derivative, which makes it special). The derivative of ln(x) is 1/x. And the derivative of √x is 1/(2√x).

Higher-Order Derivatives

You can take the derivative of a derivative. The second derivative, written as f''(x), tells you how the rate of change itself is changing. Going back to the car example, if the first derivative is speed, the second derivative is acceleration. You can explore this concept further with our acceleration calculator. The third derivative and beyond also have uses in physics and engineering.

Why Derivatives Matter

Derivatives are used to find maximum and minimum values of functions, which is useful for optimization problems. Engineers use them to design structures, economists use them to model costs and profits, and scientists use them to describe how systems change over time. In physics, derivatives are essential for understanding concepts like force, momentum, and kinetic energy. Whenever you need to understand how something is changing at a specific moment, derivatives are the tool you reach for.

Evaluating Derivatives at a Point

Once you find the derivative formula, you can plug in a specific value to get the exact slope at that point. For example, if f(x) = x³ and f'(x) = 3x², then f'(2) = 3(4) = 12. This means the original curve has a slope of 12 when x equals 2. This is especially helpful for finding tangent lines and understanding the behavior of a function at a specific location on its graph. You can verify these kinds of numerical results using our percentage calculator or check how values change with our percent change calculator. For related calculations involving how quantities change between two points, try our rate of change calculator or midpoint calculator.

Frequently Asked Questions

What functions can I type into the Derivative Calculator?

You can type polynomials like x^3 + 2x, trig functions like sin(x) or tan(x), logarithms like ln(x) or log(x), exponentials like e^x, square roots like sqrt(x), and combinations of all of these. You can also use parentheses to group terms, such as (x+1)^2 * cos(x).

How do I type multiplication in the calculator?

Use the * symbol for multiplication. For example, type 3*x or x*sin(x). The calculator also understands implicit multiplication, so typing 2x or 3sin(x) will work too.

Can I find the derivative with respect to a variable other than x?

Yes. Use the "Differentiate with respect to" dropdown menu to choose x, y, z, or t as your variable. The calculator will treat all other letters as constants.

What does the 'Evaluate at point' field do?

It lets you plug a number into the derivative to get a specific value. For example, if the derivative is 2x and you enter x=3, the calculator will return 6. You can type just a number like 3, or use the format x=3. You can also use pi or e.

What is the difference between the 1st, 2nd, and 3rd derivative?

The 1st derivative tells you the slope or rate of change of the original function. The 2nd derivative is the derivative of the 1st derivative — it tells you how the slope itself is changing. The 3rd derivative takes it one step further. Each higher order gives you more detail about how the function behaves.

How does the step-by-step solution work?

After you click Compute Derivative, the calculator shows each rule it used, such as the Power Rule, Product Rule, or Chain Rule. Each step is numbered and labeled so you can follow along and learn how the answer was found.

What is the Visual Editor tab for?

The Visual Editor lets you build a math expression by clicking the keyboard buttons below instead of typing. This can help if you are not sure how to type a function. You can then transfer the expression to the text input by clicking "Transfer to Text Input."

Does the Image Upload feature read my handwriting?

The image upload feature is currently simulated. It accepts PNG, JPG, GIF, or WebP files up to 1 MB, but it does not actually read the math from your image. For the most accurate results, type your expression directly or use the math keyboard.

What does the interactive graph show?

The graph plots two lines: the original function f(x) in blue and its derivative f'(x) in red. You can change the X Min and X Max values to zoom in or out. This helps you see how the slope of the function changes across different x values.

How does Practice Mode work?

Practice Mode gives you a random derivative problem to solve on your own. Type your answer and click Check Answer. The calculator compares your answer to the correct one. If you get stuck, click Hint to see a tip. Click New Problem to try a different question.

How do I type square roots and cube roots?

For a square root, type sqrt(x). For a cube root, type cbrt(x). You can also use the math keyboard and click the √ button, or open the Roots submenu in the full keyboard for more options like fourth roots.

Can I use inverse trig functions like arcsin or arctan?

Yes. Type arcsin(x), arccos(x), or arctan(x). You can also find these buttons in the Trigonometric submenu when you turn on the Full Pad keyboard. The calculator knows the derivatives for all six inverse trig functions.

How do I enter the constant pi or e?

Type pi for π and e for Euler's number. You can also click the π and e buttons on the full math keyboard. The calculator recognizes both as constants and treats them as numbers, not variables.

Why does my result look different from what I expected?

The calculator simplifies the derivative automatically, but its simplified form may look different from what you wrote by hand. Both answers can still be correct. For example, 2*x*1 simplifies to 2x, and x/x simplifies to 1. Check the step-by-step solution to see how the result was reached.

What happens if I enter an invalid expression?

If the calculator cannot understand your input, it will show an error message saying it could not parse the expression. Double-check that your parentheses are balanced, your operators are correct, and function names are spelled right (for example, sin not sn).

Can this calculator handle hyperbolic functions?

Yes. It supports sinh, cosh, tanh, csch, sech, and coth, as well as inverse hyperbolic functions like arsinh, arcosh, and artanh. You can find all of these in the Trigonometric submenu on the full keyboard.