Teaching
Why a calculator should show its steps
An answer you cannot reproduce is not much use in an exam. Here is why every rule is named and recorded.
Rules
Three ways people get the chain rule wrong
The missing inner derivative, the changed inside, and the half-peeled composition.
Engine
Why the answers stay in fractions
Floating point would be easier. Exact rationals are the reason the output reads like a textbook.
Rules
You rarely need the quotient rule
Every quotient is a product in disguise, and the product form is usually shorter.
Graphs
How to read a function against its derivative
Where one curve crosses zero, the other turns. Four patterns worth recognising on sight.
Rules
When to take logarithms first
The variable in the base and the exponent at once, and the long products it also rescues.
Rules
What happens on the fifth derivative
Expressions grow, patterns emerge, and the working gets long enough to need paging.
Engine
Guessing what you meant by 2x(x+1)
Implicit multiplication is convenient for the writer and awkward for the parser.
Rules
The corner where the derivative does not exist
sgn(x) is the right answer everywhere except the one point people ask about.
Engine
How do you test a derivative?
Against a finite difference, at several points, for every expression in the suite.
Rules
Why arctan has an algebraic derivative
Inverting a function reciprocates its slope, and the trigonometry cancels out.
Graphs
Choosing a window for ninety graphs
A fixed viewport ruins most curves. Two heuristics pick a sensible one automatically.