Math guides
Concept guides that pair with the calculators, learn the idea, then compute with the tool.
Each guide below is written to be read with its calculator: the guide builds intuition, the tool handles computation. Start with any topic.
All guides
Algebra, start to finish
The full pillar guide: linear equations, quadratics, slope, with worked examples and links to every tool.
Proportions, deeply
Why cross-multiplication works, read the guide on the proportion calculator page.
Quadratics, deeply
Discriminants and what they predict, on the quadratic calculator page.
Spread, deeply
Why we square deviations, on the standard deviation page.
Derivatives, deeply
The power rule from first principles, on the derivative page.
Volume formulas, deeply
Where the 1/3 comes from, across the geometry calculators.
Calculus worked examples
Derivative of xˣ: Logarithmic Differentiation
Why xˣ needs logarithmic differentiation, worked step by step.
Derivative of (x^2+1)/(x−1) by the Quotient Rule
The quotient rule applied in full, with the derivative simplified.
Implicit Differentiation x²+y²=25 (Circle)
Finding dy/dx and tangent slopes on the circle x²+y²=25.
Integral of x ln x: Integration by Parts
Integration by parts for x ln x, choosing u and dv the right way.
Integral of x²eˣ: By Parts Twice (Solved)
x²eˣ integrated by parts twice, then checked by differentiating.
Integral of eˣ sin x: The Circular Trick
The classic circular trick for ∫eˣ sin x dx, solved end to end.
Limit of (eˣ−1−x)/x² as x→0: Solved
An indeterminate limit solved with series and L’Hôpital’s rule.
Linear Approximation of sqrt(x+1) at x=3: Example
Estimating sqrt(x+1) near x = 3 with the tangent line.
Partial Derivative of x^2y^3 + e^(xy): Worked Example
Both partial derivatives of x²y³ + e^(xy), one variable at a time.
Triple Integral of xyz Over a Box: Worked Example
The triple integral of xyz over a box, one variable at a time.
Algebra worked examples
Factor x⁴−5x²+4: Biquadratic, Step by Step
A biquadratic factored by substitution, step by step.
Solve 3×3 System of Equations: Worked Example
A 3×3 system solved by elimination, with every step shown.
Solve log2(x)+log2(x−2)=3 (Extraneous Root)
A log equation whose extra root must be rejected, and why.
Solve |2x−3|=7: Absolute Value Equation
Both cases of |2x−3|=7 worked and checked.
Quadratic Equation with Complex Roots: x²+4x+5=0
Complex roots from a negative discriminant, worked in full.
Solver guides
AI Math Solver: Free Tool That Shows Every Step
What a good AI math solver does, and how to get the most from one.
Math Solver with Steps: Step-by-Step Solutions Free
How step-by-step solutions work, topic by topic.
Math Homework Solver: Solve This Question with Steps
Using a solver to learn homework methods, not just copy answers.