Calculus Solver

Enter a derivative, integral, limit, series, or common first-order differential equation. Get an exact result, readable steps, and a check you can inspect.

Runs in your browser No sign-up Exact forms first

Choose a calculus operation

Derivative Integral Limit Series Differential equation

Expression Write naturally—use the math keys for powers and fractions.

x^2\sin(x)

Variable Derivative order

Try an example:

More than calculus

Solve more complex mathwith detailed steps.

Mathos AI helps you work through harder problems across calculus, algebra, geometry, physics, and more.

Solve with Mathos AI

One workspace, five core tasks

What this calculus solver can do

The calculator selects a method from the operation you choose, keeps exact symbolic forms when possible, and tells you when a problem falls outside its supported range.

Differentiate functions

Find first, second, or third derivatives of polynomial, trigonometric, exponential, logarithmic, product, quotient, and composite expressions.

diff(sin(x^2), x)

Evaluate integrals

Compute many indefinite antiderivatives and definite integrals. Indefinite answers include the constant of integration.

integrate(x*exp(x), x)

Find limits

Evaluate finite and infinite limits symbolically, with numerical sampling used as a separate reasonableness check when it is safe.

limit(sin(x)/x, x, 0)

Build Taylor polynomials

Generate a Taylor or Maclaurin polynomial around a chosen center, using repeated derivatives up to the selected degree.

T₅ for exp(x) at 0

Solve common first-order ODEs

Handle equations of the forms y′ = f(x), y′ = f(x)y, and y′ + ay = b(x). More general ODEs return a clear limitation instead of a guessed answer.

dy/dx = x*y

How to use the calculus solver

Choose the task first, then give the calculator only the information that operation needs.

1
Choose an operation
Select Derivative, Integral, Limit, Series, or Differential equation.
2
Enter the expression
Use the visual math field, then add the variable, bounds, target point, center, or order when shown.
3
Review the result
Read the exact answer first, then inspect the method steps, conditions, and verification.

Ways to enter common notation
Expression	How to enter it
$x^{3}$	Type x, then use the power key
$3 x$	Type 3x naturally
$\frac{x + 1}{x - 1}$	Use the fraction key
$sin (x)$	Type sin(x) or use the sine key
$e^{x}$	Type e, then use the power key
$x$	Use the square-root key

Include the missing details

For a definite integral, choose Integral, enter the integrand, and provide both bounds.

\int_{0}^{1} e^{x} d x

Go to the calculator

Try a calculus example

Each example below has been tested against the calculator. Choose one to load it into the solver.

Derivatives

$\frac{d}{d x} sin (x^{2})$ Chain rule
$\frac{d}{d x} (x^{2} e^{x})$ Product rule
$\frac{d ^{2}}{d x ^{2}} (x^{3} - 4 x)$ Second derivative

Integrals

$\int x e^{x} d x$ Integration by parts
$\int_{0}^{1} e^{x} d x$ Definite integral
$\int x cos (x^{2}) d x$ Substitution

Limits

$lim_{x \to 0} \frac{s i n x}{x}$ Standard limit
$lim_{x \to 1} \frac{x ^{2} - 1}{x - 1}$ Factor and cancel

Taylor polynomials

$T_{5} [e^{x}] at x = 0$ Maclaurin polynomial
$T_{7} [sin x] at x = 0$ Odd-power terms

Differential equations

$\frac{d y}{d x} = 2 x$ Direct antiderivative
$\frac{d y}{d x} = x y$ Separable equation

A complete worked example

Find the antiderivative, explain the method choice, and verify the result.

Problem

\int x e^{x} d x

Recognize the method
The integrand is a product of an algebraic term and an exponential term, so integration by parts makes the algebraic factor simpler.
$u = x, d v = e^{x} d x$
Find $d u$ and $v$
Differentiate the part chosen as $u$ and integrate the part chosen as $d v$ .
$d u = d x, v = e^{x}$
Apply integration by parts
Substitute the four pieces into $\int u d v = uv - \int v d u$ .
$\int x e^{x} d x = x e^{x} - \int e^{x} d x$
Simplify the answer
Evaluate the remaining standard integral and include the constant of integration.
$\int x e^{x} d x = e^{x} (x - 1) + C$

Verification

Differentiating the answer returns the original integrand.

\frac{d}{d x} [e^{x} (x - 1)] = x e^{x}

Learn integration by parts

Learn the decision, not only the result

Two methods worth recognizing on sight

Differentiation

Chain rule

Use it when one differentiable function sits inside another. Learn how to identify the layers, differentiate from the outside in, and multiply the derivative factors.

Learn the chain rule

outside rate

\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)

inside rate

Integration

Integration by parts

Use it to exchange a difficult product integral for a simpler one. See where the formula comes from and how to choose the part to differentiate.

Learn integration by parts

differentiate u

\int u d v = uv - \int v d u

integrate dv

Answers you can inspect

How results are checked

A symbolic answer is useful only when you can see its assumptions and test the result. The solver uses operation-specific checks rather than displaying an unexplained confidence score.

Try a verified example

1
Parse the expression
Normalize notation, identify the variable, and reject ambiguous or unsupported input.
2
Compute an exact form
Use symbolic calculus before converting any value to a decimal approximation.
3
Run an inverse or numerical check
Differentiate an antiderivative, substitute an ODE solution, or sample a limit where the domain allows it.

Common mistakes and how to fix them

Most failed calculations come from a missing parameter, the wrong operation, or an assumption the calculator cannot infer.

Using a function operation for an equation

Fix: Choose Differential equation before entering an equation such as $\frac{d y}{d x} = 2 x$ .

Entering only one integral bound

Fix: Enter both the lower and upper bound for a definite integral, or leave both empty for an indefinite integral.

Reading a Taylor polynomial as an infinite series

Fix: Treat the result as a finite approximation around the selected center. It does not establish convergence away from that center.

Expecting a one-sided limit

Fix: This calculator currently evaluates two-sided limits. It will not infer whether you intend the left-hand or right-hand limit.

Expecting an ODE initial value to be applied

Fix: The supported ODE forms return a general solution containing $C$ . Initial conditions are not applied automatically.

Ignoring domain or branch restrictions

Fix: Review where logarithms, roots, denominators, and inverse functions are defined before using the result.

Common questions

Using the calculus calculator

What types of calculus problems can this solver solve?

It supports first through third derivatives, many indefinite and definite integrals, two-sided limits, Taylor polynomials, and documented first-order differential equations. See the supported problem types for the current boundaries.

Does the calculus solver show every step?

It shows a compact, method-focused sequence with the rule choice, important transformations, exact answer, conditions, and verification. Very small algebraic rearrangements may be combined instead of appearing as separate steps.

How do I enter derivatives, integrals, and limits?

Choose the operation first, enter the expression in the visual math field, and complete the fields that appear. For integrals, enter both bounds or neither; for limits, enter the target value. The input guide shows common notation.

How are answers checked, and how accurate are they?

The solver uses operation-specific checks, such as differentiating an antiderivative, comparing a derivative with a numerical slope, sampling a two-sided limit, or reconstructing an ODE form. These checks reduce avoidable errors but do not guarantee that every symbolic result is valid for every domain or branch.

Why does my answer look different from the textbook answer?

Two expressions can be algebraically or trigonometrically equivalent even when they look different. Indefinite integrals may also differ by a constant. Compare the expressions by simplifying their difference or, for antiderivatives, differentiating both answers.

Why did the calculator leave an expression unevaluated?

The expression may not have an elementary symbolic result, or it may be outside the supported method. The calculator reports that boundary rather than inventing a closed form.

Is the solver free, and does it save my questions?

The current calculator is free to use and does not require an account. Typed expressions are processed in this browser and are not sent to a calculation server by this implementation.

Can it handle advanced or multivariable calculus?

Not yet. This page currently focuses on single-variable calculus and the first-order ODE forms listed above. It does not claim support for partial derivatives, gradients, multiple integrals, vector calculus, or higher-order differential equations.

Calculation engine

Symbolic operations use the browser-based Nerdamer computer algebra system. See the Nerdamer documentation for supported functions.

Calculus Solver

Exact answer

How it works

Solve more complex mathwith detailed steps.

What this calculus solver can do

Differentiate functions

Evaluate integrals

Find limits

Build Taylor polynomials

Solve common first-order ODEs

How to use the calculus solver

Choose an operation

Enter the expression

Review the result

Include the missing details

Try a calculus example

Derivatives

Integrals

Limits

Taylor polynomials

Differential equations

A complete worked example

Recognize the method

Find $d u$ and $v$

Apply integration by parts

Simplify the answer

Two methods worth recognizing on sight

Chain rule

Integration by parts

How results are checked

Common mistakes and how to fix them

Using a function operation for an equation

Entering only one integral bound

Reading a Taylor polynomial as an infinite series

Expecting a one-sided limit

Expecting an ODE initial value to be applied

Ignoring domain or branch restrictions

Using the calculus calculator

Calculation engine

Calculus Solver

Exact answer

How it works

Solve more complex mathwith detailed steps.

What this calculus solver can do

Differentiate functions

Evaluate integrals

Find limits

Build Taylor polynomials

Solve common first-order ODEs

How to use the calculus solver

Choose an operation

Enter the expression

Review the result

Include the missing details

Try a calculus example

Derivatives

Integrals

Limits

Taylor polynomials

Differential equations

A complete worked example

Recognize the method

Find du and v

Apply integration by parts

Simplify the answer

Two methods worth recognizing on sight

Chain rule

Integration by parts

How results are checked

Common mistakes and how to fix them

Using a function operation for an equation

Entering only one integral bound

Reading a Taylor polynomial as an infinite series

Expecting a one-sided limit

Expecting an ODE initial value to be applied

Ignoring domain or branch restrictions

Using the calculus calculator

Calculation engine

Find $d u$ and $v$