Why is the derivative of x² equal to 2x?
A short, rigorous derivation from the limit definition, with a geometric interpretation.
Read the guide →Understand why, not just the answer.
Short, rigorous, verifiable guides for adults revisiting or deepening their mathematics.
How to factor x² − 5x + 6
The product-sum method, a direct check, and the connection to polynomial roots.
Read the guide →Bayes’ theorem: an intuitive explanation
How new evidence updates a probability without confusing P(A|B) with P(B|A).
Read the guide →Limits and the indeterminate form 0/0
Why 0/0 is not an answer and how simplification reveals a limit’s behavior.
Read the guide →Why does dividing by a fraction mean multiplying by its reciprocal?
An algebraic and intuitive explanation of the rule, with a verifiable example and the conditions under which it holds.
Read the guide →Why does any nonzero number to the zero power equal 1?
A rigorous derivation from the exponent quotient rule, with examples, a direct check, and an explanation of the 0⁰ case.
Read the guide →Why is division by zero undefined?
A rigorous explanation using the inverse operation, distinguishing a/0 from 0/0 and a function’s value from its limit.
Read the guide →How do you calculate successive percentage changes?
A rigorous multiplier method, general formula, and verifiable example for avoiding the mistake of adding percentages with different bases.
Read the guide →Why does equal-distance average speed use the harmonic mean?
A derivation of average speed over two equal-distance legs, with a 60 and 40 km/h example and a comparison with the arithmetic mean.
Read the guide →Why do independent probabilities multiply?
A derivation from conditional probability, a two-coin example, and a counterexample using draws without replacement.
Read the guide →Why does multiplying an inequality by a negative reverse the sign?
A proof using the difference between the two sides, a number-line interpretation, and a checkable example.
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