← Differentiation methods
1 Reading the notation
Calculus writes the same idea several ways. Recognising each form is half the battle.📖 Notation used in differentiation
8 termsCover the right column and check you can say what each one means.
- dy/dx
- The derivative of y with respect to x: the rate at which y changes as x changes. Read as “dee y by dee x”. It is a single symbol, not a fraction you can split (though it behaves like one in substitution).
- f′(x) (f-prime)
- The same thing in function notation. f′(x) is the derivative of f(x).
- f″(x) or d²y/dx²
- The second derivative, the derivative of the derivative. Tells you concavity, and whether a stationary point is a max or min.
- d/dx (as an operator)
- An instruction: “differentiate what follows with respect to x”. d/dx (sin x) means “differentiate sin x”.
- Δ vs d
- Capital Δ means a finite change (Δy/Δx is the gradient of a chord). Lower-case d means an infinitesimal one: the limit as the interval shrinks to zero. That limit is exactly what turns a chord gradient into a tangent gradient.
- Stationary point
- Where f′(x) = 0. The tangent is horizontal. Could be a maximum, a minimum, or a stationary point of inflection.
- Point of inflection
- Where concavity changes. Requires f″(x) = 0 AND a sign change in f″. f″ = 0 alone is not enough. Y = x⁴ has f″(0) = 0 but no inflection.
- ∝
- “Proportional to”.