Free Calculus Calculator with Step‑by‑Step Solutions

Calculus, explained: how to use this calculator and understand every step

The calculator above is a step-by-step calculus solver. Type a function, choose what you want (a derivative, an indefinite integral, a definite integral or a limit), and it returns the exact answer with every rule it used, in the order a teacher would write it. Antiderivatives are checked by differentiating them back, so you can trust the result as well as read it.

This guide explains what calculus is, how each solver mode works, how to type functions so they're read correctly, and how to check an answer yourself. It also points you to the right tool or guide for everything else, from all 48 calculators to the formula sheet.

What is calculus?

Calculus is the mathematics of change and accumulation. It has two halves that turn out to be opposites:

• Differential calculus measures how fast something changes. Its tool is the derivative: the slope of a curve at a single point, such as the speed of a car at one instant.
• Integral calculus adds up infinitely many small pieces to find a total. Its tool is the integral: the area under a curve, such as the distance a car covers from its speed.

Both rest on the idea of a limit, the value a quantity approaches as you zoom in closer and closer. The bridge between the two halves is the Fundamental Theorem of Calculus, which says that integration and differentiation undo each other:

$$\frac{d}{dx}\int_a^x f(t)\,dt = f(x), \qquad \int_a^b f'(x)\,dx = f(b) - f(a)$$

For a fuller tour of the core ideas, read calculus concepts explained. If you're preparing for a course, Calculus 1 vs Calculus 2 shows what each one covers, and the precalculus review lists the algebra and trig you'll need first.

The four solver modes

Derivative mode

The solver differentiates symbolically, the same way you would by hand, and names each rule: the power rule, product rule, quotient rule and chain rule. For \( x^3\sin x \) it applies the product rule:

$$\frac{d}{dx}\left[x^3\sin x\right] = 3x^2\sin x + x^3\cos x$$

Choose "Second derivative" for concavity and inflection points, or "Third derivative" for jerk in physics. A full table of rules is in the guide to derivative rules, and the dedicated derivative calculator page has more examples.

Integral mode

Integration is harder than differentiation because there's no single rule that always works. The solver tries the standard techniques in order: basic formulas, u-substitution, integration by parts and partial fractions. For \( \int xe^x\,dx \) it chooses integration by parts with \( u = x \) and \( dv = e^x\,dx \):

$$\int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C$$

It then differentiates the answer to confirm it gives back \( xe^x \). Some functions, like \( e^{-x^2} \), have no antiderivative made of elementary functions; the solver tells you so and suggests a definite integral instead. See integration formulas for the complete list of rules, or the integral calculator.

Definite mode

A definite integral gives a number: the signed area between the curve and the x-axis from \( a \) to \( b \). The solver finds an antiderivative when one exists and evaluates it at the bounds; otherwise it integrates numerically to high precision. Infinite bounds such as inf turn it into an improper integral:

$$\int_0^\pi \sin x\,dx = \big[-\cos x\big]_0^\pi = 1 + 1 = 2$$

Limit mode

Limits are found by direct substitution when possible, by simplifying, or by L'Hôpital's rule for indeterminate forms such as \( \frac00 \). The classic example is

$$\lim_{x\to 0}\frac{\sin x}{x} = 1$$

which is explained step by step in the limit of sin x / x guide. You can also take one-sided limits and limits at infinity; see limits at infinity and the limit calculator.

How to type functions

The preview under the input box shows how your expression was read before you solve, which catches most typing mistakes. The most common one is a missing bracket: 1/x+1 means \( \frac1x + 1 \), not \( \frac{1}{x+1} \).

A worked example from start to finish

Find the slope of \( f(x) = \frac{x^2 + 1}{x - 3} \) at \( x = 4 \).

1. Choose Derivative mode and type (x^2 + 1)/(x - 3).
2. Read the steps. The quotient rule gives \( f'(x) = \frac{2x(x - 3) - (x^2 + 1)}{(x - 3)^2} = \frac{x^2 - 6x - 1}{(x - 3)^2} \).
3. Substitute. \( f'(4) = \frac{16 - 24 - 1}{1} = -9 \).
4. Use it. The tangent line at \( (4, 17) \) is \( y - 17 = -9(x - 4) \). The tangent line calculator does this last step for you, and the equation of a tangent line guide explains it.

Mistakes the step-by-step working helps you catch

Seeing every step is what makes a solver useful for learning, because you can compare it line by line with your own work. These are the errors students make most often, and where they show up in the steps:

• Forgetting the chain rule. \( \frac{d}{dx}\sin(3x) = 3\cos(3x) \), not \( \cos(3x) \). The steps show the inner derivative being multiplied in.
• Wrong sign in the quotient rule. The numerator is \( f'g - fg' \), in that order.
• Dropping the constant of integration. Every indefinite integral ends in \( + C \), because any constant differentiates to zero.
• Treating \( \int f(x)g(x)\,dx \) as \( \int f \cdot \int g \). There's no product rule for integrals that works that way; integration by parts is the correct tool.
• Substituting into an indeterminate form. Plugging \( x = 0 \) into \( \frac{\sin x}{x} \) gives \( \frac00 \), which is not an answer; it's a signal to simplify or use L'Hôpital's rule.
• Degrees instead of radians. Calculus formulas such as \( \frac{d}{dx}\sin x = \cos x \) assume radians. Use the degrees to radians guide to convert.

How to check an answer yourself

• Differentiate your antiderivative. If \( F'(x) \) gives back the integrand, the integral is right. This is exactly how the solver verifies its own answers.
• Plug in a number. Compare \( f'(a) \) with the slope between two nearby points, \( \frac{f(a + 0.001) - f(a)}{0.001} \).
• Graph it. A derivative should be zero where the original graph has a peak or valley. Plot both in the graphing calculator.
• Check units and signs. Areas below the x-axis count as negative in a definite integral; distances can't be negative.

Where calculus is used

Motion is the classic application: velocity and acceleration are derivatives of position, and the kinematic equations come from integrating constant acceleration. Change that compounds, like population growth or radioactive decay, leads to differential equations.

Every tool and guide, in one place

The solver above handles most problems, but some questions need a specialized tool. Each has its own page with instructions, a worked example and answers to common questions:

• All calculators: 48 tools grouped by topic, including critical points, Riemann sums, Taylor series, partial derivatives and projectile motion.
• Graphing calculator: plot any function with its derivative, tangent line and shaded area.
• Practice problems: questions with instant worked explanations.
• Formula sheet: searchable derivatives, integrals, limits and series formulas, plus free printable PDF formula sheets for calculus, trig, algebra, geometry and statistics.
• Algebra and trig tools for the skills calculus relies on: the quadratic formula calculator, trig calculator, logarithm calculator and scientific calculator.
• Geometry, statistics and everyday maths: the area, volume, Pythagorean theorem, standard deviation, z-score, matrix, percentage, fraction and compound interest calculators.

For learning a topic properly, the guides go further than any calculator. Start with derivatives of trig functions, u-substitution, the unit circle or trig identities, or browse the blog by category. Want a full textbook? See the free, open-licensed calculus textbooks and the official NCERT maths books for Class 11 and 12.

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