How do I type functions?

Use ^ for powers, sqrt() for roots, e^x or exp(x), ln(x) for the natural log, log(x) for base 10, |x| or abs(x), and pi. Multiplication can be implicit: 2x, x sin(x), 3(x+1).

Why does it say no elementary rule found?

Some integrals, like ∫e^(x²) dx, have no elementary antiderivative, and others need partial fractions or trigonometric substitution, which are not built in yet. Definite integrals are still computed numerically.

Which differential equations can it solve?

Separable and linear first-order equations, y′ = f(x), and second-order linear equations with constant coefficients. Anything else is solved numerically when you give initial values.

Can I trust the answers?

Integrals are verified by differentiating them back, and differential-equation solutions are checked by substituting them into the equation. If a check fails, the result is not shown.

Calculus & Differential Equations Calculator

Derivatives, integrals, limits, Taylor series and differential equations solved step by step, with every rule explained and a graph.

DifferentiateType a function of x: x^2, sin(x), e^x, ln(x), sqrt(x), |x|

f(x)Reads as x²e^x

Order

Examples

Derivativef′(x) = x²e^x + 2xe^x

Product Ruled/dx[x² · e^x] = d/dx[x²] · e^x + x² · d/dx[e^x](fg)′ = f′g + fg′
Power Ruled/dx[x²] = 2xd/dx[xⁿ] = n·xⁿ⁻¹
Exponential Ruled/dx[e^x] = e^xd/dx[eᵘ] = eᵘ·u′
Simplifyd/dx[x²e^x] = x²e^x + 2xe^x

Graphf(x) · f′(x) (dashed)

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What Is the Calculus & Differential Equations Calculator?

A step-by-step calculus solver. Derivatives of any order name each rule (power, product, quotient, chain, exponential, logarithmic differentiation). Integrals use table rules, linear substitution, u-substitution, integration by parts and expansion, show every step, add the constant, and evaluate definite integrals with the Fundamental Theorem of Calculus (or numerically when there is no elementary antiderivative). Limits try direct substitution, then L’Hôpital’s rule for 0/0 and ∞/∞, then a numerical check, from both sides or one side, including limits at infinity. Taylor and Maclaurin series list each derivative. Differential equations cover separable, linear first-order and direct-integration equations, and second-order linear equations with constant coefficients (characteristic equation; undetermined coefficients for polynomial, exponential, sine and cosine forcing), with initial values, a numerical RK4 solution and a slope field.

How It Works

Choose Derivative, Integral, Limit, Taylor Series or Differential Equation, type the function (x^2 e^x, sin(x)/x, y'' + 3y' + 2y = 0) and set the options: derivative order, integral bounds, the limit point and side, the series center and order, or initial values like y(0)=1, y'(0)=0. The answer, the numbered working and a graph update as you type.

When to Use It

Use it to check homework, learn which rule applies and why, or verify a solution before an exam. Each integral is checked by differentiating it back before it is shown.

Frequently Asked Questions

How do I type functions?: Use ^ for powers, sqrt() for roots, e^x or exp(x), ln(x) for the natural log, log(x) for base 10, |x| or abs(x), and pi. Multiplication can be implicit: 2x, x sin(x), 3(x+1).
Why does it say no elementary rule found?: Some integrals, like ∫e^(x²) dx, have no elementary antiderivative, and others need partial fractions or trigonometric substitution, which are not built in yet. Definite integrals are still computed numerically.
Which differential equations can it solve?: Separable and linear first-order equations, y′ = f(x), and second-order linear equations with constant coefficients. Anything else is solved numerically when you give initial values.
Can I trust the answers?: Integrals are verified by differentiating them back, and differential-equation solutions are checked by substituting them into the equation. If a check fails, the result is not shown.

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