Math Calculators - Derivatives, Integrals, Limits, Vectors, Factoring & More

Exponential & Log Equation Solver

$ a^x = b \Rightarrow x = \dfrac{\ln b}{\ln a},\; \log_b(x)=c \Rightarrow x = b^c $

Solve exponential and logarithmic equations step-by-step using inverse functions and log rules.

Asymptote Finder

$ x = a \text{ (vertical)},\; y = b \text{ (horizontal)},\; y = mx + b \text{ (oblique)} $

Find vertical, horizontal, and oblique (slant) asymptotes of rational functions with step-by-step limits.

Work & Energy

$ W=\vec{F}\cdot\vec{d},\quad E_k=\tfrac{1}{2}mv^2,\quad E_p=mgh $

Compute work with angle between force and displacement; kinetic and potential energies.

Double Integral Calculator (Rectangular Region)

$ \iint_R f(x,y)\,dA = \int_a^b \int_c^d f(x,y)\,dy\,dx $

Evaluate double integrals over rectangular regions with step-by-step inner and outer integration.

Parallel & Perpendicular Lines

$ m_{\parallel}=m,\quad m_{\perp}=-\dfrac{1}{m} $

Given a line and a point, find the line that is parallel or perpendicular through that point.

Vector Magnitude

$ \|\vec{v}\| = \sqrt{\sum_{i=1}^{n} v_i^2} $

Compute the length of a vector with exact and decimal forms.

Unit Circle Values

$ (\cos\theta,\sin\theta)\text{ on }r=1 $

Exact sin/cos/tan for special angles with quadrant signs.

Allele Frequency from Genotype Counts

$ N=n_{AA}+n_{Aa}+n_{aa},\; p=\dfrac{2n_{AA}+n_{Aa}}{2N},\; q=1-p $

Convert genotype counts to allele frequencies (p, q) quickly.

Derivative Calculator

Enter a function and get the derivative with rules used at each step. Supports power, product, quotient, and chain rules.

Ideal Gas Law

$ PV = nRT $

Solve for P, V, n, or T using PV=nRT with consistent units.

Chi-Square Test Calculator

$ \chi^2=\sum \dfrac{(O_i-E_i)^2}{E_i} $

Run chi-square goodness-of-fit and independence tests with observed and expected counts, plus step-by-step tables.

Section Formula (Internal Division)

$ P\,(x,y) = \left(\dfrac{m x_2 + n x_1}{m+n},\, \dfrac{m y_2 + n y_1}{m+n}\right) $

Find the point that divides a segment internally in a given ratio m:n.

Confidence Interval (Mean/Proportion)

$ \text{CI} = \hat{\theta} \pm z_{\alpha/2}\,\text{SE}(\hat{\theta}) $

Build z/t confidence intervals for a mean (σ known/unknown) or a proportion with clear standard-errors.

Standard Deviation & Variance

$ s = \sqrt{\frac{\sum_{i=1}^n (x_i-\bar{x})^2}{n-1}},\quad \sigma = \sqrt{\frac{\sum_{i=1}^n (x_i-\mu)^2}{n}} $

Compute variance and standard deviation step-by-step. Choose sample (n−1) or population (n) denominator. Shows mean, deviations, squared deviations, and exact/decimal results.

Kinematics (SUVAT)

$ s=ut+\tfrac{1}{2}at^2,\; v=u+at,\; v^2=u^2+2as $

Compute displacement, velocity, or time with constant-acceleration formulas.

Midpoint of a Segment

$ M\,(x_m,y_m) = \left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right) $

Find the midpoint between two points with step-by-step simplification.

Ohm’s Law (V, I, R)

$ V = I\,R $

Solve for voltage, current, or resistance given the other two.

Correlation Coefficient (Pearson r)

$ r = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2\sum (y_i-\bar{y})^2}} $

Compute strength and direction of linear association between two variables.

Matrix Multiplication Calculator

$ (AB)_{ij} = \sum_{k=1}^n a_{ik}b_{kj} $

Multiply two matrices step-by-step, showing each row-by-column dot product.

Projectile Motion

$ x=v_0\cos\theta\,t,\; y=v_0\sin\theta\,t-\tfrac{1}{2}gt^2 $

Horizontal/vertical components, flight time, and range on level ground.

Matrix Inverse Calculator

$ A^{-1} = \dfrac{1}{\det(A)}\,\operatorname{adj}(A) $

Compute the inverse of a matrix with determinant/adjugate steps.

Distance Between Two Points

$ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} $

Compute the distance between two points with the classic formula.

Hardy–Weinberg Equilibrium

$ p^2 + 2pq + q^2 = 1,\quad p+q=1 $

Compute allele frequencies (p, q), expected genotype frequencies (p^2, 2pq, q^2), and a chi-square goodness-of-fit test for HWE.

Fraction Calculator

$ \frac{a}{b}\;\pm\;\frac{c}{d}=\frac{ad\pm bc}{bd} $

Add, subtract, multiply, or divide fractions with step-by-step work. Includes simplifying and mixed number answers for Grade 5-6.

Partial Derivative Calculator

$ \frac{\partial f}{\partial x},\; \frac{\partial f}{\partial y} $

Compute first-order partial derivatives of multivariable functions with step-by-step differentiation rules.

Rational Equation Solver

$ \frac{2}{x-1}+\frac{3}{x+2}=\frac{5x+1}{x^2+x-2} $

Solve equations with rational expressions; check extraneous roots.

Molar Mass (Chemical Formula)

$ M=\sum_i n_i M_i $

Parse a chemical formula and sum atomic masses (periodic-table values).

Series/Parallel Resistance

$ R_{\text{series}}=\sum R_i,\quad \frac{1}{R_{\text{parallel}}}=\sum \frac{1}{R_i} $

Compute equivalent resistance for series or parallel networks.

Poisson Distribution Calculator

$ P(X=k)=\dfrac{\lambda^k e^{-\lambda}}{k!} $

Compute Poisson probabilities P(X=k), cumulative probabilities, and expected counts with step-by-step work.

Determinant (2×2 / 3×3)

$ \det(A)\text{ via }ad-bc\text{ or cofactor expansion} $

Compute det(A) for 2×2/3×3 with steps (minors/cofactors).

System of Equations Solver $(2×2 / 3×3)$

$ \begin{cases} 2x+3y=7 \\ -x+4y=5 \end{cases} $

Solve linear systems with real numbers. Shows elimination or substitution steps and the final numeric solution.

Law of Sines

$ \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} $

Solve triangles using a/sin A = b/sin B = c/sin C. Handles ASA/AAS and SSA (ambiguous case) with full steps.

One-Way ANOVA Calculator

$ F=\dfrac{\text{MS}_B}{\text{MS}_W} = \dfrac{\text{SS}_B/(k-1)}{\text{SS}_W/(N-k)} $

Perform one-way ANOVA from raw data or group summaries, reporting F-statistic, df, p-value, and an ANOVA table.

Polynomial Long Division Calculator

$ f(x)=q(x)g(x)+r(x) $

Divide one polynomial by another using long division, with each step of the algorithm clearly shown.

Point–Slope → Slope–Intercept

$ y - y_1 = m(x - x_1) \;\to\; y = mx + b $

Convert y − y1 = m(x − x1) into y = mx + b with clear steps.

Taylor/Maclaurin Series

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

Expand common functions around a point; show first N terms.

Stoichiometry (Moles ↔ Mass)

$ n=\frac{m}{M}\; $

Convert given mass to moles, apply balanced-equation mole ratios, convert to product mass.

Z-Test Calculator

$ z=\dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}} $

Perform one-sample and two-sample z-tests with known σ. Get z-statistic, p-value, and a clear conclusion.

Gradient & Directional Derivative Calculator

$ \nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle,\; D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u} $

Find the gradient vector and directional derivative at a point in the direction of a given vector, with full steps.

Domain & Range Calculator

$ f: x \mapsto f(x) $

Find the domain and range of a function, explaining restrictions, asymptotes, and key values.

Angle Between Vectors

$ \cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\,\|\vec{b}\|} $

Compute the angle using dot product and norms.

Sequence Rule Finder

$ a_n = a_1 + (n-1)d \text{ or } a_n = a_1 r^{n-1} $

Detect whether a sequence is arithmetic or geometric and find its explicit formula with step-by-step reasoning.

Mean / Median / Mode

$ \bar{x}=\frac{1}{n}\sum_{i=1}^{n} x_i,\quad \text{median}=\text{middle value},\quad \text{mode}=\text{most frequent} $

Compute measures of center: mean, median, and mode(s). Handles ties and even/odd n.

Dot Product

$ \vec{a}\cdot\vec{b} = \sum_{i=1}^{n} a_i b_i = \|\vec{a}\|\,\|\vec{b}\|\cos\theta $

Compute a·b and the angle between vectors.

Triangle Solver (ASA/SSS/SAS/SSA)

$ a/\sin A = b/\sin B = c/\sin C,\quad c^2=a^2+b^2-2ab\cos C $

Solve any triangle (ASA, SSS, SAS, SSA). Returns all sides/angles with steps.

Matrix Rank

$ \operatorname{rank}(A)=\text{number of pivots in }\operatorname{rref}(A) $

Compute rank via rref and show pivot columns.

Cross Product (3D)

$ \vec{a}\times\vec{b} = \begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_1&a_2&a_3\\ b_1&b_2&b_3\end{vmatrix} $

Compute a×b and its magnitude with steps.

t-Test Calculator

$ t=\dfrac{\bar{x}-\mu_0}{s/\sqrt{n}} $

Perform one-sample and two-sample t-tests. Get t-statistic, degrees of freedom, and p-value with clear steps.

Eigenvalues & Eigenvectors

$ \det(A-\lambda I)=0,\quad (A-\lambda I)\,\mathbf{v}=\mathbf{0} $

Find eigenvalues/eigenvectors for 2×2 or 3×3 matrices.

Logarithm Simplifier

$ \log_b(xy)=\log_b x + \log_b y $

Apply product/quotient/power rules and change of base.

Covariance Calculator

$ \operatorname{Cov}(X,Y)=\dfrac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y}) $

Compute sample covariance step-by-step from paired (x, y) data, with intermediate tables.

Factoring Polynomials

Factor quadratics and higher-degree polynomials: grouping, rational root test, and special products.

Permutations & Combinations

$ {}^{n}P_{r}=\frac{n!}{(n-r)!}\\{}^{n}C_{r}=\frac{n!}{r!(n-r)!} $

Compute ordered selections (nPr) and unordered selections (nCr).

Limit Calculator

$ \lim_{x\to a} f(x) $

Evaluate limits with a step-by-step math AI solver. Handles rational, trigonometric, and indeterminate forms with L'Hôpital’s rule.

Solution Dilution (M₁V₁ = M₂V₂)

$ M_1V_1=M_2V_2 $

Solve for any one of M1, V1, M2, or V2 given the other three.

Law of Cosines

$ c^2=a^2+b^2-2ab\cos C $

Find sides/angles from SAS or SSS using the Law of Cosines.

Probability Rules & Bayes' Theorem

$ P(A\cup B)=P(A)+P(B)-P(A\cap B),\; P(A\mid B)=\dfrac{P(A\cap B)}{P(B)},\; P(A\mid B)=\dfrac{P(B\mid A)P(A)}{P(B)} $

Compute unions, intersections, conditional probabilities, and Bayes' theorem with step-by-step explanations.

Definite Integral Calculator

Evaluate definite integrals step-by-step with substitutions, part-by-part evaluation, and the Fundamental Theorem of Calculus.

Linear Regression (y = a + bx)

$ b = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2} $

Fit y = a + bx, report slope/intercept, and predict at new x.

Integral Calculator

Solve integrals with step-by-step substitutions, parts, trig identities, partial fractions, and numeric checks.

Orthonormal Basis (Gram–Schmidt)

$ u_1=\dfrac{v_1}{\|v_1\|},\quad u_2=\dfrac{v_2-\operatorname{proj}_{u_1}v_2}{\|v_2-\operatorname{proj}_{u_1}v_2\|},\;\dots\quad\ \text{where } \\ \mathrm{proj}_{u}w=(w\cdot u)u $

Orthonormalize a set of vectors with step-by-step projections.

Partial Fractions Decomposition

Decompose rational functions (distinct, repeated, irreducible quadratics) with a clean coefficient solve.

Degrees ↔ Radians

$ 180^\circ = \pi\ \text{radians} $

Convert angles between degrees and radians with exact \pi forms.

Z-Score Calculator

$ z = \frac{x-\mu}{\sigma} $

Standardize a value and return its normal percentile.

Binomial Probability

$ P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} $

Exact P(X=k) for X~Bin(n,p).

Quadratic Equation Solver

Solve $ax^2+bx+c=0$ by factoring, completing the square, or quadratic formula with discriminant analysis and graph insights.

RREF (Row-Reduced Echelon Form)

$ R\sim I\text{ via Gaussian/Jordan elimination} $

Row-reduce a matrix to echelon/rref with pivot operations shown.

Slope–Intercept $(y = mx + b)$

$ y = mx + b $

Find slope and intercept from two points or an equation. Shows step-by-step to y=mx+b.

Standard ↔ Vertex Form

$ ax^2+bx+c \;\leftrightarrow\; a(x-h)^2 + k $

Complete the square to convert ax^2+bx+c into a(x−h)^2+k and back.

Trigonometric Identity Verifier

$ \sin^2 x + \cos^2 x = 1 $

Verify identities or simplify trigonometric expressions.

Hypergeometric Distribution Calculator

$ P(X=k)=\dfrac{{K \choose k}{N-K \choose n-k}}{{N \choose n}} $

Compute exact hypergeometric probabilities for draws without replacement, with full step-by-step work.

Distance from Point to Line

$ d = \dfrac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}} \quad (Ax+By+C=0) $

Compute perpendicular distance from a point to a line Ax+By+C=0.

Vector Projection

$ \operatorname{proj}_{\vec{b}}\vec{a} = \dfrac{\vec{a}\cdot\vec{b}}{\|\vec{b}\|^2}\,\vec{b} $

Compute projection of a onto b and the perpendicular component.

Standard ↔ Slope–Intercept

$ Ax + By = C \;\leftrightarrow\; y = mx + b $

Convert Ax+By=C to y=mx+b or the reverse with step-by-step algebra.

Point/Line/Plane (3D) — Distances & Intersections

$ d_{\text{pt→plane}}=\dfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}} $

Distance point→plane, line→plane intersection point, and parallel detection.

Arithmetic & Geometric Series

$ S_n=\frac{n}{2}(a_1+a_n),\quad S_n=a_1\,\frac{1-r^n}{1-r} $

Sum arithmetic and geometric series with numeric examples.

Rational Function Analyzer

$ f(x)=\dfrac{P(x)}{Q(x)} $

Analyze rational functions: domain, vertical/horizontal/oblique asymptotes, holes, and intercepts with full steps.

Discriminant (Δ = b² − 4ac)

$ \Delta = b^2 - 4ac $

Compute Δ=b²−4ac and interpret the number of real roots.

Normal Distribution (PDF/CDF)

$ f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/(2\sigma^2)} $

Evaluate the normal pdf or cdf at x; standardize to Z as needed.

LU Decomposition Calculator

$ A = LU $

Factor a square matrix A into L and U using Gaussian elimination with step-by-step row operations.

Line Through Two Points

$ m = \dfrac{y_2-y_1}{x_2-x_1},\; y - y_1 = m(x - x_1)\;\to\; y=mx+b $

Build the line through two points: slope, point–slope, and slope–intercept.

Synthetic Division Calculator

$ f(x)=(x-c)q(x)+f(c) $

Use synthetic division to divide by x − c, evaluate f(c), and find quotient and remainder, with table steps.