Differentiation and Applications
The Differentiation and Applications domain tests technique and interpretation: differentiating standard, composite, implicit and parametric functions, then using the derivative to describe a curve or a changing quantity. Marks are split between the mechanics of the derivative and the reasoning that follows from it, so a correct derivative with no conclusion still loses half the credit.
Core rules
- d/dx(x^n) = nx^(n-1); d/dx(sin x) = cos x; d/dx(cos x) = -sin x; d/dx(tan x) = sec^2 x; d/dx(ln x) = 1/x; d/dx(e^x) = e^x.
- Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v - uv')/v^2. Chain: dy/dx = (dy/du)(du/dx).
- Parametric: dy/dx = (dy/dt)/(dx/dt). Implicit: differentiate term by term, treating y as a function of x and attaching dy/dx.
- Stationary points solve dy/dx = 0; classify by the sign change of dy/dx or by y'' (y'' < 0 maximum, y'' > 0 minimum, y'' = 0 inconclusive).
- Tangent at (x1, y1): y - y1 = m(x - x1) with m = dy/dx there; the normal has gradient -1/m.
Worked example
For y = x^3 - 3x^2 + 4, dy/dx = 3x^2 - 6x = 3x(x - 2), so stationary points occur at x = 0 and x = 2. Since y'' = 6x - 6, at x = 0 we get y'' = -6 < 0, a maximum with y = 4; at x = 2 we get y'' = 6 > 0, a minimum with y = 0.
Traps
- Dropping the inner derivative in a chain-rule application such as sin 3x or e^(2x).
- Assuming y'' = 0 signals a point of inflexion; check the sign of dy/dx on both sides instead.
- Failing to test the endpoints when a greatest or least value is required on a closed interval.
- Using the reciprocal rather than the negative reciprocal for a normal, or substituting the wrong coordinate into the tangent equation.