www.calculuscalc.comInteractive Calculus Problem Solver27 Authentic Gadgets

Master complex calculus through rigorous steps & live visuals

CalculusCalc bridges algebraic problem solving with geometric intuition. Solve derivatives, integrals, and limits with step-by-step working, and experiment with interactive 2D simulations.

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Practice problems
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Formulas
Fundamental Theorem of Calculus
\[\frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)\]

The rate at which accumulated area grows at \(x\) is exactly the height of the curve there.

  • Symbolic derivatives with rulesd/dx
  • Verified antiderivatives∫ dx
  • Limits incl. L'Hôpitallim
  • Improper integrals to ∞GK-15

Step-by-step solverExact symbolic engine

Interactive Calculus Problem Solver

Derivatives, antiderivatives, definite and improper integrals, and limits. Every answer comes with the rules used, and antiderivatives are verified by differentiating them back.

Try:
Input syntax
  • Powers x^2, roots sqrt(x), cbrt(x), absolute value |x|
  • Implicit multiplication works: 3x sin(2x)
  • sin cos tan sec csc cot, asin acos atan, sinh cosh tanh
  • e^x or exp(x); ln(x) and log(x) are both the natural log
  • Constants pi and e; bounds accept inf and -inf
Enter a function and press Solve to see a full worked solution.

27 calculus gadgetsEvery one computes live

Specialized Calculators & Engineering Gadgets

Tangent lines, extrema, Newton's method, arc length, solids of revolution, Taylor polynomials, series tests, gradients, double integrals, Euler's method and more. Edit any input and press the button (or Enter).

Basic & Scientific#1

Simple Calculator

Everyday arithmetic with brackets, percentages and a running history. Tap the keys or type.

STD
History
    Full guide & calculator
    Basic & Scientific#2

    Scientific Calculator

    Trig in degrees or radians, inverse trig, ln and log₁₀, powers, roots, factorials, π, e and scientific notation.

    History
      Full guide & calculator
      Differential#3

      Symbolic Derivative Engine

      Exact \(f'(x)\) and \(f''(x)\) with simplification, evaluated at any point.

      Differential#4

      Tangent & Normal Line Generator

      Slope \(m = f'(x_0)\) plus the equations of the tangent and normal lines.

      Differential#5

      Critical Points & Extrema Finder

      Solves \(f'(x) = 0\), classifies each point, and finds absolute extrema on \([a, b]\).

      Differential#6

      Inflection & Concavity Analyzer

      Locates sign changes of \(f''(x)\) and lists intervals of concavity.

      Full guide & calculator
      Differential#7

      Newton's Method Root Finder

      Iterates \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) and shows every step.

      Differential#8

      Linear Approximation (Linearization)

      \(L(x) = f(a) + f'(a)(x - a)\) with the true error of the estimate.

      Differential#9

      Mean Value Theorem Solver

      Finds every \(c \in (a, b)\) with \(f'(c) = \frac{f(b) - f(a)}{b - a}\).

      Integral#10

      Definite & Improper Integral

      \(\int_a^b f(x)\,dx\) with exact antiderivatives when possible. Bounds accept inf and pi.

      Integral#11

      Area Between Two Curves

      \(A = \int_a^b |f(x) - g(x)|\,dx\), split automatically at intersections.

      Integral#13

      Solid of Revolution (Disk Method)

      Rotation about the x-axis: \(V = \pi \int_a^b [f(x)]^2\,dx\)

      Integral#14

      Solid of Revolution (Shell Method)

      Rotation about the y-axis: \(V = 2\pi \int_a^b x\,f(x)\,dx\)

      Integral#15

      Average Value of a Function

      \(f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx\)

      Integral#16

      Riemann Sum Approximator

      Left, right, midpoint, trapezoid and Simpson sums compared with the exact value.

      Series & Limits#17

      Limit Evaluator

      Two-sided and one-sided limits, including \(x \to \pm\infty\).

      Series & Limits#18

      L'Hôpital's Rule Evaluator

      Resolves \(\tfrac{0}{0}\) and \(\tfrac{\infty}{\infty}\) forms by differentiating top and bottom.

      Series & Limits#19

      Taylor Polynomial Generator

      Degree-\(n\) Taylor/Maclaurin polynomial centered at \(a\), with error check.

      Full guide & calculator
      Series & Limits#20

      Series Convergence Tester

      Partial sums of \(\sum a_n\) with the ratio test and \(n\)th-term test. Use n as the variable.

      Multivariable#21

      Partial Derivatives

      First and second partials of \(f(x, y)\), evaluated at a point.

      Multivariable#22

      Gradient & Directional Derivative

      \(\nabla f\) and \(D_{\mathbf{u}} f = \nabla f \cdot \hat{\mathbf{u}}\) at a point.

      Full guide & calculator
      Multivariable#23

      Double Integral over a Rectangle

      \(\iint_R f(x, y)\,dA\) for \(R = [a, b] \times [c, d]\).

      Physics & Applied#24

      Euler's Method ODE Solver

      Approximates \(y' = F(x, y)\) step by step, compared with RK4.

      Physics & Applied#25

      Motion: Position → Velocity → Acceleration

      \(v(t) = s'(t)\), \(a(t) = s''(t)\), and whether the object is speeding up. Use t.

      Physics & Applied#26

      Work Done by a Variable Force

      \(W = \int_a^b F(x)\,dx\) (newtons × metres = joules).

      Physics & Applied#27

      Exponential Growth & Decay

      Solves \(\frac{dP}{dt} = kP\): \(P(t) = P_0 e^{kt}\), with doubling time or half-life.

      2D Visual LabLive coordinate sandbox

      See derivatives and integrals take shape

      Plot any function, slide a tangent line along the curve, shade the signed area under it, and watch Riemann rectangles converge to the exact integral.

      f(x) f′(x) Tangent line Signed area Drag to pan · Ctrl/⌘ + scroll to zoom

      Practice arenaAP & university level

      Test yourself, one problem at a time

      Pick a topic, answer, and get an instant worked explanation. Questions are shuffled every round.

      0 / 0 correct

      Formula cheatsheet44 essentials

      Every formula you reach for, in one place

      Search by name or topic. Tap TeX to copy the LaTeX source.

      Download printable PDF

      9Derivatives

      Power rule

      \[\frac{d}{dx}x^n = n x^{n-1}\]

      Product rule

      \[(fg)' = f'g + fg'\]

      Quotient rule

      \[\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}\]

      Chain rule

      \[\frac{d}{dx} f(g(x)) = f'(g(x))\, g'(x)\]

      Sine & cosine

      \[\begin{gathered}(\sin x)' = \cos x \\ (\cos x)' = -\sin x\end{gathered}\]

      Tangent & secant

      \[\begin{gathered}(\tan x)' = \sec^2 x \\ (\sec x)' = \sec x \tan x\end{gathered}\]

      Exponentials

      \[\begin{gathered}(e^x)' = e^x \\ (a^x)' = a^x \ln a\end{gathered}\]

      Natural log

      \[(\ln x)' = \frac{1}{x}\]

      Inverse trig

      \[\begin{gathered}(\arcsin x)' = \frac{1}{\sqrt{1 - x^2}} \\ (\arctan x)' = \frac{1}{1 + x^2}\end{gathered}\]

      9Integrals

      Power rule

      \[\begin{gathered}\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \\ n \neq -1\end{gathered}\]

      Reciprocal

      \[\int \frac{1}{x}\,dx = \ln|x| + C\]

      Exponentials

      \[\begin{gathered}\int e^x\,dx = e^x + C \\ \int a^x\,dx = \frac{a^x}{\ln a} + C\end{gathered}\]

      Sine & cosine

      \[\begin{gathered}\int \sin x\,dx = -\cos x + C \\ \int \cos x\,dx = \sin x + C\end{gathered}\]

      Secant squared

      \[\int \sec^2 x\,dx = \tan x + C\]

      Arctangent form

      \[\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\arctan\frac{x}{a} + C\]

      Arcsine form

      \[\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin\frac{x}{a} + C\]

      Integration by parts

      \[\int u\,dv = uv - \int v\,du\]

      u-substitution

      \[\int f(g(x))\,g'(x)\,dx = \int f(u)\,du\]

      6Limits

      Sine limit

      \[\lim_{x \to 0} \frac{\sin x}{x} = 1\]

      Cosine limit

      \[\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\]

      Exponential limit

      \[\lim_{x \to 0} \frac{e^x - 1}{x} = 1\]

      Definition of e

      \[\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e\]

      L'Hôpital's rule

      \[\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}\ \text{for } \tfrac{0}{0}, \tfrac{\infty}{\infty}\]

      Derivative definition

      \[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]

      8Series

      Geometric series

      \[\begin{gathered}\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r} \\ |r| < 1\end{gathered}\]

      p-series

      \[\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges} \iff p > 1\]

      Ratio test

      \[\begin{gathered}L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| \\ L < 1 \text{ converges},\ L > 1 \text{ diverges}\end{gathered}\]

      Taylor series

      \[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n\]

      Exponential

      \[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]

      Sine

      \[\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\]

      Cosine

      \[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\]

      Natural log

      \[\begin{gathered}\ln(1 + x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} \\ -1 < x \le 1\end{gathered}\]

      5Theorems

      Fundamental Theorem I

      \[\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\]

      Fundamental Theorem II

      \[\int_a^b f(x)\,dx = F(b) - F(a)\]

      Mean Value Theorem

      \[f'(c) = \frac{f(b) - f(a)}{b - a},\ c \in (a, b)\]

      Rolle's Theorem

      \[\begin{gathered}f(a) = f(b) \\ \exists\, c \in (a, b): f'(c) = 0\end{gathered}\]

      Intermediate Value Thm.

      \[\begin{gathered}f(a) < N < f(b) \\ \exists\, c \in (a, b): f(c) = N\end{gathered}\]

      7Applications

      Arc length

      \[L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx\]

      Disk method

      \[V = \pi \int_a^b [f(x)]^2\,dx\]

      Washer method

      \[V = \pi \int_a^b \left(R(x)^2 - r(x)^2\right)dx\]

      Shell method

      \[V = 2\pi \int_a^b x\,f(x)\,dx\]

      Average value

      \[f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx\]

      Newton's method

      \[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

      Linear approximation

      \[L(x) = f(a) + f'(a)(x - a)\]