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Calculus solver

Solve a calculus problem

Enter one complete task, then review the interpreted expression, exact answer, conditions, named steps, and verification.

Use the math keyboard or your device keyboard.
Try an example

Result

Product-rule derivative

2xsin⁡(x)+x2cos⁡(x)2x\sin(x)+x^2\cos(x)
Conditions
  • x is real.
  • The expression is differentiable for every real x.

Steps

  1. Recognize a product. The function multiplies x squared by sine of x.(fg)′=f′g+fg′(fg)'=f'g+fg'
  2. Differentiate both factors. The derivatives are 2x and cosine of x.(x2)′=2x,(sin⁡x)′=cos⁡x(x^2)'=2x,\qquad (\sin x)'=\cos x
  3. Substitute into the rule. Keep each untouched factor beside the derivative of the other.2xsin⁡(x)+x2cos⁡(x)2x\sin(x)+x^2\cos(x)
Independent check

Numerical central-difference slopes agree with the symbolic derivative at x = 0.5, 1, and 2 within the chosen tolerance.

Scope

What this calculus solver covers

Use this calculus solver for supported derivative, integral, and limit problems. Calculations run in the browser, and the surrounding explanation stays visible without requiring an account.

Derivatives

Differentiate algebraic, exponential, logarithmic, and trigonometric combinations with respect to a stated variable.

Examples: product rule, chain rule, implicit differentiation

Integrals

Find supported antiderivatives and evaluate definite integrals when bounds are supplied.

Examples: power rule, substitution, integration by parts

Limits

Evaluate supported finite, infinite, and one-sided limits while preserving the direction in the result.

Examples: removable forms, standard trigonometric limits, limits at infinity

Series and approximations

Review verified Taylor and Maclaurin examples. Automatic series expansion is outside this calculator's supported scope.

Examples: Maclaurin polynomial examples

Differential equations

Review direct-antiderivative and simple separable examples. General differential-equation solving is outside this calculator's supported scope.

Examples: direct antiderivatives, simple separable forms

Applied calculus

Static examples connect symbolic work to tangent lines, area, and optimization without pretending to parse every word problem.

Examples: tangent slope, signed area, maximum value

How to use it

Enter enough information for one clear task

  1. 1
    Name the operation

    Write differentiate, integrate, or limit, or select the matching operation.

  2. 2
    Enter the full expression

    Use parentheses to preserve grouping and include the variable, bounds, approach point, or direction.

  3. 3
    Confirm the interpreted problem

    Before trusting the answer, make sure the displayed operation matches what you intended.

  4. 4
    Read the check

    Use the named steps and independent verification to compare the result with your own work.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Derivatives

Derivative

product rule

ddx(x2ex)\frac{d}{dx}(x^2e^x)

Expected result

ex(x2+2x)e^x(x^2+2x)

Derivative

chain rule

ddxsin⁡(x2)\frac{d}{dx}\sin(x^2)

Expected result

2xcos⁡(x2)2x\cos(x^2)
Show more derivatives examples

Implicit derivative

implicit differentiation

x2+y2=25x^2+y^2=25

Expected result

dydx=−xy\frac{dy}{dx}=-\frac{x}{y}

Integrals

Integral

substitution

∫xex2 dx\int xe^{x^2}\,dx

Expected result

12ex2+C\frac12e^{x^2}+C

Definite integral

endpoint evaluation

∫0πsin⁡(x) dx\int_0^{\pi}\sin(x)\,dx

Expected result

22
Show more integrals examples

Integral

integration by parts

∫xln⁡(x) dx\int x\ln(x)\,dx

Expected result

x22ln⁡(x)−x24+C\frac{x^2}{2}\ln(x)-\frac{x^2}{4}+C

Limits

Limit

standard limit

lim⁡x→0sin⁡xx\lim_{x\to0}\frac{\sin x}{x}

Expected result

11

One-sided limit

sign analysis

lim⁡x→0+x∣x∣\lim_{x\to0^+}\frac{x}{|x|}

Expected result

11
Show more limits examples

Limit

standard exponential limit

lim⁡x→0ex−1x\lim_{x\to0}\frac{e^x-1}{x}

Expected result

11

Series and applications

Taylor polynomial

Maclaurin expansion

P5(x) for sin⁡xP_5(x)\text{ for }\sin x

Expected result

x−x36+x5120x-\frac{x^3}{6}+\frac{x^5}{120}

Tangent line

point-slope form

f(x)=x3−2x at x=1f(x)=x^3-2x\text{ at }x=1

Expected result

y=x−2y=x-2
Show more series and applications examples

Optimization

single-variable maximum

2L+2W=202L+2W=20

Expected result

L=W=5,Amax⁡=25L=W=5,\quad A_{\max}=25

Complete example

Evaluate an integral by substitution

The exponent contains x squared, and its derivative is proportional to the remaining factor x. That structure points to substitution.

∫xex2 dx\int xe^{x^2}\,dx
  1. 1
    Choose the inside expression

    Let u equal x squared so the exponential becomes e to the u.

    u=x2,du=2x dxu=x^2,\qquad du=2x\,dx
  2. 2
    Replace x dx

    Solve for the factor that appears in the integral.

    x dx=12dux\,dx=\frac12du
  3. 3
    Integrate in u

    The transformed integral is a constant multiple of the exponential antiderivative.

    12∫eu du=12eu+C\frac12\int e^u\,du=\frac12e^u+C
  4. 4
    Substitute back

    Replace u with the original inside expression.

    12ex2+C\frac12e^{x^2}+C
∫xex2 dx=12ex2+C\int xe^{x^2}\,dx=\frac12e^{x^2}+C

Verification: Differentiating one half e to the x squared gives one half times e to the x squared times 2x, which simplifies to the original integrand.

Avoidable errors

Common mistakes and how to fix them

Missing grouping

Problem: Typing sin x^2 when the intended function is the square of sine.

Why it matters: The expressions sine of x squared and sine squared of x have different derivatives.

Better approach: Use parentheses or rendered notation: sin(x^2) versus (sin(x))^2.

Missing variable

Problem: Asking to differentiate an expression with x and y without naming the variable.

Why it matters: A partial derivative and an implicit derivative are different tasks.

Better approach: State the differentiation variable and whether other symbols are constants or dependent variables.

Lost bounds

Problem: Entering a definite integral as an indefinite one.

Why it matters: An antiderivative family and an accumulated value answer different questions.

Better approach: Include both lower and upper bounds and preserve them in the interpreted problem.

Lost limit direction

Problem: Omitting the plus or minus direction at a jump.

Why it matters: The left and right limits can disagree even when each exists.

Better approach: Choose left, right, or two-sided explicitly when the function changes behavior at the point.

Premature decimals

Problem: Rounding constants in the first step.

Why it matters: Small rounding errors can grow and can hide an exact cancellation.

Better approach: Keep exact fractions and constants until a decimal is actually requested.

Unverified equivalence

Problem: Rejecting a result only because it looks different from a textbook form.

Why it matters: Correct antiderivatives and simplified expressions can have multiple forms.

Better approach: Differentiate, simplify the difference, and compare domains before deciding.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • Simplify the result independently and compare equivalent forms.
  • Substitute regular sample values when a numerical check is meaningful.
  • Check the domain and any one-sided, endpoint, or constant assumptions.

When to stop and revise the input

  • Revise an expression when the variable, boundary, direction, or grouping is ambiguous.
  • Treat a symbolic result as unsupported when the engine cannot establish an elementary form.
  • Use the steps to check work, not as a substitute for course or examination rules.

Common questions

Solve a calculus problem FAQ

Does the calculator show steps?

Yes. A result includes the interpreted problem, the final answer, the mathematical operation used, relevant conditions, and a verification note when an independent check is available.

Why can a correct answer look different from a textbook answer?

Equivalent algebraic and trigonometric forms can look different while representing the same function. Indefinite integrals can also differ by a constant. Compare by simplifying, differentiating, or substituting values within the shared domain.

What should I include in the input?

State the expression, variable, and operation. Add an interval, upper and lower bounds, a one-sided direction, derivative order, or initial condition whenever the task depends on it.

How should I check an important result?

Inspect the recognized problem and assumptions first. Then use an independent check such as differentiating an antiderivative, substituting a solution into the original equation, or comparing one-sided numerical values.

Continue learning

Choose the resource that matches what you need to do next.

Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.