Derivative Calculator • With Complete Steps and Solution

Calculate the Derivative of…

Enter any function below. Supports sin, cos, tan, ln, e^x, polynomials, products, quotients and more.

Calculate the Derivative of …

Use parentheses as needed. See or change variable & order in .

This will be calculated:

Enter a function above…

Result

Enter the function above. Change variable and order in . Click Go! to compute.

The Derivative Calculator lets you calculate derivatives of functions online — for free!

Our calculator shows you the full working (step-by-step differentiation) so you can check your solutions and learn.

Supports 1st through 5th order derivatives, partial derivatives, and all common functions.

Happy differentiating!

How to Use

Type your function in the input field
Use toolbar buttons to insert operators
Set variable and order in Options
Click Go! or press Enter

Input Format

x^2 → x²
sin(x) → sin(x)
exp(x) or e^x → eˣ
log(x) → ln(x)
sqrt(x) → √x
* for multiplication
pi → π, e → e

Key Rules

Power: d/dx[xⁿ] = n·xⁿ⁻¹
Product: (fg)’ = f’g + fg’
Quotient: (f/g)’ = (f’g−fg’)/g²
Chain: f(g(x))’ = f'(g)·g’

Click an example to load it:

Differentiation Variable Derivative Order

Current Settings Variable: x
Order: 1st derivative

Solve and check your answer!

Find the derivative of:

Your answer f'(x) =

What Is a Derivative Calculator?

A derivative calculator is an online tool that helps you quickly find the derivative of any mathematical function. In simple words, it makes difficult calculations easy by solving them in seconds. It not only gives the final answer but also shows step-by-step solutions, so you can understand the process clearly and improve your skills.

For example, if you enter x², the calculator instantly gives 2x as the result. This tool is very helpful for students working on assignments, teachers explaining concepts, and beginners who want to learn calculus in a simple and practical way.

Quick Examples

f(x) = x²→f′(x) = 2xPower

f(x) = sin(x)→f′(x) = cos(x)Trig

f(x) = eˣ→f′(x) = eˣExp

f(x) = ln(x)→f′(x) = 1/xLog

How to Use the Derivative Calculator

Using our derivative calculator is simple and quick. Just follow these easy steps.

1

⌨️

Enter Your Function

Type your math function: x², sin(x), e^x or (x²+1)/x. You can also use the virtual keyboard.

2

🔤

Select the Variable

Usually x, but you can change to y, t, or z from the dropdown menu.

3

🔢

Choose Order

Pick 1st for regular derivative, 2nd for second derivative, or higher.

4

▶️

Click Compute

Press the Compute button or Enter key. Result appears instantly.

5

✅

Read Step-by-Step

Click “Show Steps” to see which rules were applied and how the answer was reached.

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Tip: You can write 2x without the × sign — the calculator understands implicit multiplication automatically.

Key Features

A high-quality derivative calculator gives users both quick results and better understanding.

Step-by-Step Solutions

Shows which rule (Power, Chain, Product, Quotient) was applied at every stage.

Instant Results

Works immediately, saving time during homework or practice sessions.

All Function Types

Algebraic, trigonometric, exponential, and logarithmic expressions all supported.

Interactive Graph

See f(x) and f′(x) side by side on a real-time chart — zoom with the slider.

Higher-Order Derivatives

Calculate 1st through 5th order derivatives in one click.

Free & Always Available

No sign-up required — use it anytime, anywhere, on any device.

Types of Derivatives You Can Calculate

Our derivative calculator supports all major types of derivatives used in calculus.

First Derivative

Basic

Shows how fast a function is changing at a specific point — tells you the slope.

d/dx[x³] = 3x²

Second Derivative

Intermediate

Explains how the rate of change itself changes — used to measure concavity.

d²/dx²[x³] = 6x

Higher-Order Derivatives

Advanced

Go beyond the second derivative — used in advanced physics and engineering.

d³/dx³[x⁴] = 24x

Partial Derivatives

Multivariable

Differentiate functions with multiple variables — keep other variables constant.

∂/∂x[x²y] = 2xy

Implicit Differentiation

Special

Used when variables are mixed together and cannot be separated easily — finds derivatives without isolating y.

d/dx[x² + y²] = 2x + 2y·(dy/dx)

Common Derivative Rules

These rules form the foundation of almost every derivative problem.

Rule	Formula	Description
Constant Rule	`d/dx[c] = 0`	Derivative of any constant is always 0.
Power Rule	`d/dx[xⁿ] = n·xⁿ⁻¹`	Bring the power down and reduce exponent by one.
Sum/Difference Rule	`d/dx[f±g] = f′ ± g′`	Differentiate each term separately.
Product Rule	`d/dx[f·g] = f′g + fg′`	Used when two functions are multiplied together.
Quotient Rule	`d/dx[f/g] = (f′g − fg′)/g²`	Applied when one function is divided by another.
Chain Rule	`d/dx[f(g(x))] = f′(g(x))·g′(x)`	For functions nested inside other functions.

Note: Our calculator automatically applies all these rules — the Steps section shows you exactly which rule was used.

Worked Examples

Detailed examples showing how each derivative rule is applied in practice.

Example 1 — Power Rule

d/dx[x⁵]

▶Bring exponent down, reduce by 1: 5·x⁵⁻¹

Example 2 — Trig Rule

d/dx[sin(x)]

▶The derivative of sin is cos — a fundamental trig rule.

Example 3 — Exponential

d/dx[eˣ]

▶eˣ is unique — its derivative is itself.

Example 4 — Product Rule

d/dx[sin(x)·eˣ]

▶Apply f′g + fg′: cos(x)eˣ + sin(x)eˣ

Example 5 — Chain Rule

d/dx[sin(x²)]

▶Outer: cos(x²). Multiply by inner derivative 2x.

Example 6 — Quotient Rule

d/dx[(x²+1)/(x−1)]

▶Apply (f′g − fg′)/g² then simplify.

Benefits of Using an Online Derivative Calculator

Why use an online derivative calculator instead of solving by hand?

✓

Saves Time

Solves problems in seconds instead of minutes of manual work.

✓

Reduces Errors

Far more accurate than manual calculations — no human mistakes.

✓

Improves Understanding

Step-by-step solutions build real knowledge and conceptual clarity.

✓

Easy to Use

Simple interface anyone can navigate immediately without training.

✓

Always Available

Access anytime, anywhere, on any device for free.

✓

Supports All Levels

Perfect for beginners and advanced learners alike.

Common Mistakes in Derivatives

Small mistakes can lead to wrong answers. Try to avoid these common errors.

⚠️

Missing the Chain Rule

Forgetting to multiply by the inner derivative when differentiating composite functions like sin(x²).

⚠️

Incorrect Signs (+/−)

Sign errors are extremely common and can flip the entire answer.

⚠️

Skipping Steps

Rushing through the solution leads to errors in the middle steps.

⚠️

Wrong Bracket Placement

Not using brackets properly changes the order of operations and the result.

⚠️

Mixing Up Rules

Confusing the product rule with the quotient rule or chain rule.

Always: Review your work carefully. Use our calculator to verify your manual results.

Real-Life Applications of Derivatives

Derivatives are not just used in math — they play an important role in many real-life fields.

⚡

Physics

Measure speed, motion, and acceleration of objects.

💰

Economics

Analyze profit, cost, and market changes.

⚙️

Engineering

Design and optimize systems and structures.

🌿

Biology

Study growth rates and population dynamics.

Derivative Calculator vs Manual Solving

Both methods have advantages — using them together gives the best results.

Feature	Calculator	Manual
Speed	Instant — results in seconds	Takes more time
Accuracy	Highly accurate, no human error	Can have calculation errors
Learning	Step-by-step solutions help learn	Builds deeper understanding
Availability	Anytime, anywhere, free	Requires pen, paper, and focus

Best approach: Use the calculator for quick answers and manual solving to strengthen your concepts.

FAQs About Derivative Calculator

Everything you need to know about using the derivative calculator.

A derivative calculator is an online tool that quickly computes derivatives of mathematical functions. It shows step-by-step solutions identifying which rules (Power, Chain, Product, Quotient) were applied.

Yes, this derivative calculator is completely free. No sign-up, no limits. Use it anytime you need.

Yes! After computing, click “Show Step-by-Step Solution” to see exactly which derivative rules were applied and why — perfect for learning.

The derivative of x² is 2x. Using the Power Rule: bring the exponent (2) down as a coefficient and reduce the power by one — giving 2·x¹ = 2x.

The derivative of sin(x) is cos(x). This is a fundamental trigonometric derivative rule. If the argument is a function of x (e.g. sin(x²)), the Chain Rule also applies: 2x·cos(x²).

Yes — select 2nd, 3rd, 4th, or 5th from the Order dropdown. The calculator applies differentiation repeatedly the selected number of times.

All common functions: polynomials (x^3), trig (sin, cos, tan, cot, sec, csc, arcsin, arccos, arctan), exponential (e^x), logarithmic (ln, log), square root (sqrt), and absolute value (abs) — plus any combination using +, −, ×, ÷, ^.

Yes! The calculator automatically handles implicit multiplication. You can write 2x, 3x², x(x+1), etc. — it will correctly interpret them as 2*x, 3*x^2, x*(x+1).

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Final Thoughts

A derivative calculator is a very helpful tool for anyone learning calculus. It turns complex problems into simple steps and makes understanding much easier.

Stay consistent, keep practicing, and you will master derivatives step by step.